(1) (2) (3) (4) 1 - PDF Free Download

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3.................................... 3........................ 6.3 B [, ].......................... 8.4........................... 9........................................... 9................................. 44.3................................... 53 3 85 3.......................................... 85 3.................................... 86 3.3................................... 94 3.4................................ 95 3.5.................................. 3.6 ψ.................................... 6 4 5 4. NECDET BATIR......................... 8 A A.................................... A............................

. B B B B B B B B d B B. B. B B tdt + B. B. B k. B B B d * B k k.3 k + B d B tdt + B + B. B, B [ ] + B + B. k B k k k B + B B..3 * C k k 3

.3. + k B + + + + + k k B tdt + B + k k B k t k + B + k B k [ k + +! k!k! B + + k k k ] t k+ + B k + + B k k k + k+ + B +. B k + B k k+ + B + k k +.3 B + + B k + k. k k + +! k +!k! k + k+ + B + + k B d [B ] B B. B B B..4.3.4 B k B k.5 k.5 B, B 6, B 3 B, B, B 6, B 3, B 4 3, B 5, B 6 4, B 7, B 8 3, B 9, B 5 66, B, B 69 73, B 3, B 4 7 6, B 5, B 6 367 5, B 7, B 8 43867 798,... 4

. B, B t, B + 6, B 3 3 3 +, B 4 4 3 + 3, B 5 5 5 4 + 5 3 3 6, B 6 6 3 5 + 5 4 + 4, B 7 7 7 6 + 7 5 7 6 3 + 6, B 8 8 4 7 + 4 3 6 7 3 4 + 3 3, B 9 9 9 8 + 6 7 5 5 + 3 3, B 5 9 + 5 8 7 6 + 5 4 3 + 5 66, y B y B +, I B.6 B B.6 B B II k.6 B k k B k k + f B k+ k+ B k+ f B k+ k+ B k+ f C k + B k k+ k + B k k + [ B k k B k ] fd fd B k+ d k+ B k+ d B k+ s ds s B k+ sds. Cd C C. k +.6 5

B B B B.4 B + B +.7 B + B + B +..8 B.8 B B.9 B. B + B. I B.. B + B + [II] k. B k + B k k k. k + f B k+ + B k+ f B k+ + B k+ k + {B k + B k } k + k k. f k + k k d k + k + C. C B k+ B k+ C..4 B B B C. f B k+ + B k+ k + k k +.... 6

q B + B + q + B + + q q + + B q q B q. B. I B q B + k q k q k. + k q + q qq II. + q f B + k + q k q f + B + k + k q q + B + k q k + q [ q k B + k q f C q + B + q q B q q + B + q q + B ] q B fd q fd k q q k.. q B + + k q q k+ q k q B + sds B + sds. Cd C C. B d q B + d + k q s + B + B. 7

. B + B B B B..3.3 B [, ] 3 B [, ] B B B y B [, ] B [, ],, y B B [, ],, y B α, β < α < < β I B + 6, B B 6, B B [, ],, y B 3 B 3 3B B [, ] s, t < s < < t < s t B 3 + + B 3 α 3 s, β 3 t II k k, k + k + B k+ k + B k+, k + y B k+ 8

B k+ α k+ β k+ + + B k+ B k+ α k+ β k+ + B k+ y B K+ B k+ + B k+ B k+, \ B k+ B k+ B k+ B k+ d > B k+ d B k+ B k+ , B k+ B k+ <. 9

4 B k+ + B k+ B k+ <, B k+ B k+ >. 3 B k+ [, ] s k+, t t+ < s t+ < < t t+ < B k+3 s k+ t k+ + B k+3 α k+3 s k+, β k+3 t k+ 4 B [, ] B B B B.3 B B.4 B B < B B [, ] B B B y.5 y B 3 y B 7 O.5.5

y.5 y B 8 y B 4 O.5.5 y.5 y B 9 y B 5 O.5.5 y.5 y B O.5 y B 6.5

.4. f [a, b] +. N h b a N N fa + jh h j + b a k θ < θ < fd + [fa + fb] B [ ] k k! hk f k b f k a + B N + +! h+ f + a + θh + jh j B B. 4 g [, ] + gtdt g + g + + k + +! g + g + k B + +! B k k! B k k! g k g k B + tg + tdt.5 g k g k { B } +t g + tdt.6 B +.5.6!!.5.6. I.

gtdt B tgtdt [B tgt] B tg tdt, g + g g + g g + g B tg tdt [ B tg t + B! ] B t t, B t B t + g g B tg tdt, B 3t 3B t [ ] + 3! B 3tg t 3! B 3tg tdt B 3 B 3, B 4t 4B 3 t g + g g + g g + g + g + g + B! + B! + B! g g 4! B 4tg tdt [ g g 4! B 4tg t g g + B 4 4! 4! B 4tg 4 tdt + k B k k! ] + g 3 g 3 g k g k + 4! 4! B 4tg 4 tdt B 4 tg 4 tdt..5 +! B + tg + tdt + 3! [ ] + 3! B +3tg + t B + 4! +4tg +3 tdt [ ] + 4! B +4tg +3 t + B +4 + 4! g +3 g +3 B +3tg + tdt + 3! + 4! + + 4! B +3 tg +3 tdt B +4 tg +4 tdt B +4 tg +4 tdt. 3

gtdt g + g + g + g + B +4 + 4! g + g + + + k +! + + k B k k! B k k! g k g k B + tg + tdt g k g k g +3 g +3 + + k + 4! B k k! +.5.5 gtdt g + g + + B + +! R B + +! B + +! +! B + +! k B k k! + + 4! g k g k B +4 tg +4 tdt g k g k g + g + g + g + +.6 g + tdt + +! + +! {B + B + t} g + tdt { B } +t g + tdt B + +! B +4 tg +4 tdt B + tg + tdt. B + tg + tdt B + tg + tdt 4

F [, N] + N F j j N θ < θ < 4 g F + j, F t + jdt t + j s j+ j F d + [F + F N] + k B k k! [ ] F k N F k + B N + F + θ + j.7 +! F tdt j F j + F j + j,,..., N + B + +! F t + jdt F j + F j + j+ j + B + +! j,,..., N k B k k! F k j F k j + { B +t B + F sds k B k k! } F + t + jdt. F k j F k j + { B +t B + } F + t + jdt. N j j+ j F tdt N j F j + F j + B + +! + k B k k! N j { B +t B + F k j F k j + } N j F + t + jdt. i N N j G F tdt N [F + F N] + B + +! j F k + F + + j N j k { B +t B + B k k! } N [ ] F k f k N j F + t + jdt. F + + j [, ] G m, M. 5

wt B +t B + < t < wt > 3 B [, ] B B B, < t [, ] G m M m G M. w> mw wg Mw. m wd < wgd < M wd> m < G wd / wgd wd < M. θ Gθ wgd wd, < θ < wd R B + +! B + +! d B + +! Gθ B + B + d { B } +t Gtdt B + wtgtdt B N + F + θ + j. +! j 6

[F + F N] N F tdt + N [F + F N] F k +.7 ii G N j j F + + j R B + +! B +Gt +! k B k k! B N + F + θ + j. +! B + +! Gt B + +! Gθ j { B +t B + { B +t B + B N + F + θ + j. +! j [ ] F k k N } Gtdt } dt θ < θ <. [F + F N] N F tdt + N [F + F N] F k + j k B k k! B N + F + θ + j. +! j [ ] F k k N.7 [, N] [a, b]. [, N] [a, b], t a + ht, h b a N 7

f [a, b] +. N h b a N h N fa + jh j b a fd + [fa + fb] + k B [ ] k k! hk f k b f k a + B N + +! h+ f + a + θh + jh.8 j θ < θ < F fa + h N fa + jh j N h fa + hd + b a + k B k k! [fa + fa + Nh] [ ] h f k a + Nh h f k a + B N + +! h+ f + a + hθ + j j fsds + [fa + fb] B [ ] + k k! h f k b f k a k + h + B + +! N j f + a + θh + jh. a + h s.8. 8

... f a, b PCF. a, b f + lim f, f lim f +. f PCF. a, b fa +. lim f, fb lim f a+ b a, b f [a, b]. a, b. P CF. fa fa + lim f, fb fb lim f fa, fb f a+ b [a, b] [a, b]. a, b f,,..., m a < < < < m < m b. a, b m,,,,..., m, m. i, i+ f i +, f i+ i+ i f d. m m f d + f d + + f d f d m. b a f d, f [, ] s f.. f d +s +s f d f f + f. 9

f, [ + s, + s]. +s fd. t fd + +s +s +s fd +s fd +s ft + dt +s ftdt. +s fd fd + +s fd +s +s fd +s fd +s fd... f f + f f lim +. f. f lim f f f f lim + f f, lim + +. f f R f f lim + +. f f L lim f f

. [, ] f. < < f < < f R, f L, f R, f L, f R, f L f +, f, f +, f, f +, f,...3 m, si m cos d, cos m cos d cos m cos d { m \ si m si d m, m \ si m si d m.

..4. + cos k k si + si \ m, m, ±, ±,...,. si + si d... si.. si + cos k si + cos k si k k si + k cos k + si + + si si + si + + si cos k si k. cos k. si + si d [ + cos k d + k k si k k ]..

. I si + d,,... I + I. si I + I [ si si + cos si cos d ] si si si d + d I + I. I d I I I...5. θ si θ θ,.3 θ si θ θ..4 i θ θ si θ θ. fθ θ si θ, gθ θ + si θ f θ cos θ, g θ + cos θ fθ, gθ θ fθ f, gθ g. θ θ θ si θ θ θ si θ θ ii F θ si θ θ θ F θ cos θ. F θ θ θ ξ. 3

ξ / F F F θ. 3 y θ y O - y θ y si θ 3 θ.4 θ F θ si θ < θ F θ θ. F θ θ cos θ si θ Gθ θ cos θ si θ G θ θ y y si θ θ si θ <. Gθ < θ y θ lim G θ Gθ < F θ <. θ + F θ < θ F θ G O θ θ si θ.. e +,.5 log >..6 i f e + f e. f. f + f 4

f f f e +. ii g log > g. g. f + f g g g log. y y e y + y y log O 3. e > + +! + 3 3! + +! >..7. i f e + f e. > f > [, f. 5

f > f > f e > +. ii m m.7 e > + +! g e + +! + 3 3! + 3 3! + + m m! + + m m! g e + +! > * + m+ m +! + 3 3! + + m. m! * > g > [, g. g > g > g e > + +! + 3 3! m +.7. + + m m! i ii.7. + m+ m +!. 4 α. lim lim e,.8 log,.9 lim log,. + lim + α log.. i >.7 e > + k k k! > + +!. lim ii i lim t +! t e t. lim < +! < e log lim lim e. e t log 6 lim t. log t t e t.

iii t lim log lim + t t log log t lim t t t. iv t α lim + α log lim + α α log α α lim t log t. t +..6 3 a, b] f a α f α < α a f d. lim a+ aα f lim a+ aα f a α f. a, b] f lim a+ aα f α < α a f d. 7

6 [a, f lim α f α α > a f d...7 I {f }. f I f I ϵ > N N f f < ϵ. N ϵ N Nϵ,. N ϵ f I {f } I f ϵ > N I N f f < ϵ. I f f. {f } I I ϵ > N m, N f m f < ϵ. {f } I ϵ > N I m, N f m f < ϵ. 4 I f,, 3,... f f I. 5 [a, b] f,, 3,... f. f d a a f d. 8

6 [a, b] f,, 3,... f f {f } f f...8 f,, 3,... I f + f + + f + f. I : S f + f + + f f k {S }. k f lim S f I f I ϵ > N N. f f k < ϵ k N ϵ N Nϵ,. N ϵ f I f I f ϵ > N I N. f f k < ϵ k 9

f I I ϵ > N m >, N m f k < ϵ k+. f I ϵ > N I m >, N m f k < ϵ k+. 7 I f M f I M 8 I f,, 3,... f I f f I. 9 [a, b] f,, 3,... f [a, b] f a b ft dt a a f dt. I f,, 3,... f I f f I d d f d d f. 3

..9 K : a b, α α α f, α. f, α K α α. g [a, b] ϵ > δ > α α < δ, α \ α f, α g < ϵ. δ ϵ δ δϵ,. δ ϵ f, α α α K g f, α α α K g ϵ > δ > [a, b] α α < δ, α \ α α f, α g < ϵ. α K : a b, α α α f, α α K g ϵ > R > [a, b] α > R α f, α g < ϵ. K : a b, α α α f, α, α b a α f, α d α α α α. b b α f, α d dα f, α dα d. α a a α 3 f α, α K d dα b a f, α d b K : c, α α α f, α F α, t t F α c c a f, α d f α, α d. f, α d α F α ϵ > R > α [α, α ] t > R t fα, t g < ϵ. 3

K : c, α α α f, α, α F α F α α α c c c f, α d f, α d α α α α. f, α d dα c α 3 f α, α K F α d dα c α f, α dα d. c f, α d f, α d c f α, α d. 3 α α α f, α M c f, α d α α α. c M d.. a + a + a + + a + a. < R. > R. R. 4 α\. < α.. 5. β. > β. 6. R R lim a. 7. R. a a + 3

R lim a a + 8 a R >.. f a, < R a R, R R, R [ ρ, ρ] ρ < R.f R, R. b R, R f. c < R f a a R. d < R. a f d a + + a + + R a, b a b + a b + a b + + a b a b + a b + + a b 9 < R mir, R a, b R, R a b + a b + + a b a b + a b + + a b a b. a R f a, < R 33

f R, R.. a R > R f. a lim f fr a R. R.. s > s. s s > y s > y s s < y s > S N N N s < + d s + s N s < + s. {S N } N. s s. y > S N N N+ > N + s >. s ζs d logn +. +. s ζs. ζs s s >. 34

.. a + + 3 + + log,,... {a }. y > y y < y > k k + O k k + k k+ k + < k d < k.3. + + 3 + + > + d log +. a + + 3 + + log > log + >. 35

.3 + + < d log + log. a a + log + log + >. {a } lim a γ. γ. γ lim + + 3 + + log. 36

..3 [a, b] f f b lim a f cos d, b lim a f si d..4 f a, b b lim a f cos d, b lim a f si d..5 a, b fa +, fb fa fa, fb fb f [a, b]. f [a, b] ϵ δ ξ < δ f fξ < ϵ b a. [a, b] N a,,,..., N b. b a N < δ N. f [a, b] f M M. b a f cos d N k N k N k < N N f cos d k N k f f k cos d + f k cos d k k k N k f f k d + f k cos d k k k k k k ϵ b a k k + N M si k si k ϵ b a b a N + N M ϵ + MN. 37

4MN ϵ MN < ϵ b f cos d < ϵ. a b lim a f cos d. f m µ, µ,..., µ m µ a < µ < µ < < µ m < µ m b. [a, b] m [µ, µ ], [µ, µ ],..., [µ m, µ m ]. [µ k, µ k+ ] µk+ lim f cos d. µ k lim b a f cos d lim b lim a m k µk+ f si d µ k f cos d.. 3, f α β α + β. f cos d,,,.... f si d,,... S N α N + α cos + β si {f S N } d {f S N } d 3. {f} d fs N d + {S N } d 38

{ 3 fs N d {S N } α f α α + { α α 4 α 4 { α } N + α cos + β si d α f d + N {α f cos d + β N α α + β β } N + α + β. N + α cos + β si } { N N + α α cos + β si + α cos + β si N + α α cos + β si + } f si d d } N α cos + β si + α i α j cos i cos j + β i β j si i si j + i<j i,j α i α j cos i si j α N + α 4 + β { } α N + α + β. {f S N } d. A {f} d { } N + α + β α {f} d α N + α + β..6 N α + β {A N }. α + β. 39

3.6N. {f} d α + α + β.7 3 α α + β, β α + β α + β α, β. lim α, 4 lim β lim α, lim β., g a f cos d,,,..., b f si d,,... lim a, lim b 3. g < < f g+ g < < f, f, f cos α f cos d lim a lim α. f cos d g cos d a. g < < f g < < f, f, 4

f si β f si d lim b lim β. 5 f si d, g f si d b. lim g si + d. 4. g si + d g si si cos + cos si d g cos si d + g si cos d. g cos, g si, 4 lim g cos si d, lim g si + d. lim g si cos d. 6 f, f R lim f si + si d f+. 5. f si + d {f f+} si + d si si + f+ si + d. si 4

. si + d si. {f f+} si + si d f f+ si si + d g f f+ si lim g lim + + f f+ si f R. g, 5 lim lim g si {f f+} + si d. + si d. lim f si + si d f+...4 si p < q q p si d si q p d, si si d. d. d. ϵ ϵ < p < q [ cos d cos ] q p 4 q p cos d cos p p cos q q q p cos d,

q p si d p + q q + p d p + [ q + ] q p p < ϵ. si + si d..8 si d. f si <, f f f. f. lim f lim si si lim + + si + si lim + lim + f. cos lim si + cos + si + cos + cos si si si + cos + cos [, ] lim f cos + d. lim si + d lim si + si d..8 lim si + d. t + lim si + + d lim si t t dt si t t dt si t t dt. 43

si d cos cos d d cos si d. d [ cos ] +. si d si d cos d. cos si t si d.. si t t dt... [, ] f f f, f a b f cos d,,...,.9 f si d,,.... a, b f a, b a + a cos + b si f f a + a cos + b si. 44

. f f + f, f + f f f... 7, f, f f L f R f + + f a + a cos + b si.. f f a + a cos + b si..3 6. a S N S N a N + a cos + b si f cos d, b fs ds + N { + fs ds + N f si d,,... fs cos s ds cos } fs si s ds si fs cos s cos + si s si ds fs ds + N fs cos s ds { N } fs + cos s ds. 45

{ } fs N + cos s,, S N { } + fs N + + cos s ds { N } fs + + cos s ds + { } + fs N + cos s ds { } f u N + cos u du u s + { } f + u N + cos u du u s { } f u N + cos u du + { } f + u N + cos u du f u si N + u si u du + f + u si N + u si u N + u si f u si u du. gu f u gu, g R lim u + 6. lim gu g+ u f u si f u f lim u + u N + u si u f L du g+ f du. lim f + u si N + u si u du f + 46

lim S N f + f + {f + f + }.. f. f f + + f. Dii. S N f u si N + u du + f + u si N + u du si u si u si N + u du si u S N f. {f u + f + u f} f u + f + u f u si u si u φ u f + u + f u f N + u si u si du N + u du. φ u u [, ] φ u u u si u [, ] lim N f u + f + u f u u si u si N + u du. 47

lim N {S N f}. f f. 8, f, f u + f + u f u du f f. f a + a cos + b si...3 9 [, ] f f f f, f [, ] f. f + f, f,. f f R f L f. 7 f. f a + a cos + b si. f, 3 α β α + β. a f cos d { [ ] si f β, f cos d,,,..., f si d,,... f si 48 d f si d } f si d

b f si d { [ ] cos f + { cos f + cos α. α + β + α α + β f cos } f cos d f + f f cos d f f α + β + β d cos } d α + β + a cos + b si a + b α + β α + β. 3 α + β +. [, ] f f f f,.f f a + a cos + b si f a si + b cos. f f + f. f f + f. f. f + f f f. f f R f L f. 7 f f f α + α cos + β si. 49

. α β f cos d,,,.... f si d,,... 9 α b, β a. α f 3 f d [f] f f a b cos a si + a cos + b si., f +, f f f. F f a + a cos + b si ft dt ft dt a + + F [, ]. F F. 3 F f a 7, a si b cos + + ft dt + ft dt ft dt a +,. F α + α cos + β si 5

.. α β F cos d,,,... F si d,,... α F cos d { [ ] F si b, f si d + a β F si d { [ ] F cos +. a F si } F si d f cos d a α F. si d F cos F cos cos d } d d d F α + α cos α + α { f a } si d { f a } cos d α + α. ft dt a + + α + α cos + β si a + + a + + [ α { cos + + } + β si ] [ b { cos + + } + a si ] 5

a d + a cos d + b si d f a + a cos + b si a. 5

.3 f f + f,..3. f, y O f si b f si d. f cos. a f d d [ 3 3 ] 3. a f cos d cos d { [ ] si } si d 4 4 cos d 4 { [ ] cos } cos d 4 { [ ] } cos si 4. si d si d 53

3 4 cos.4 cos + cos 3 3 + cos +..4.4 + + 3 + + + 6..5.5.6 + 3 4 + + +..6 + 3 + 5 + + + 8..7.3. < < f < < y O 3 4, f cos a f cos d. f si. 54

b f si d { [ cos { [ si ] } ] si d } d cos cos d., < < si..8 t t si si t < <..8 < t <. t si t < t <. t si < <..9 744..9 t 4. 3 + 5 7 + + + 4.3.3 3 3 f < < 55

a, b f si d ] cos + { [ { [ si ] si d } cos d } cos d., si < < si < <..3.3.8. si < <. f,.3. t dt si t dt. 6 4 { cos }..4. 4 t 4 dt cos cos t dt. 56

3 si 3..3 3 3 3 3 + 5 3 + 3 + 3 3..3 ζ3.3..3.4 4 4 f cos λ, b, a f cos d cos λ cos d {cosλ + + cosλ } d [ siλ + + λ + siλ + + λ + siλ λ siλ λ ]. siλ ± si λ cos ± cos λ si si λ cos si λ a si λ λ + + λ si λ λ λ. cos λ si λ λ + λ si λ λ cos. cos λ si λ λ + λ si λ λ. 57

λ cos si + si. si cot + + + + \, ±, ±,.....33 t cot t t + t t t + t +. t + t cot + + +..34 + ±, ±,.... < d d log si log si si.33 cot cot d [ log ] cot + si log d. log + + +. <..35 < < g + + g +. < <, g + + g < < 58

.33 si + + + +..36 +..37.3.5 5 5 f si λ, a, b f si d [ siλ + λ + siλ + λ + si λ λ. siλ λ siλ λ si λ si λ si d ] si λ {cosλ + cosλ } d si λ λ + λ λ si. si λ si λ λ si. si λ si λ cos λ cos λ 4 λ. z sec z 4 4z..38 59

z sec 4 4..39.3.6 6 6 a >. f { cosh a + cosh a e p cos q d e p si q d ep p p cos q + q si q + C,.4 + q ep p p si q q cos q + C..4 + q b, a f cos d e a + e a cos d } {e a e a cos + e a e a cos d { [ ] [ e a e a a a cos + si + e a e a + a a cos si + { } e a a + ae a cos + a + e a a + ae a cos a aea e a a +. cosh a ea e a a + a a cos. + ] } e a + e a ea e a a + a a +. a a + ea + e a e a e a a. 6

a a + a + a a + ea + e a e a e a. a + ea + e a ae a e a.4. a,.3.7 7 7 a <. f + e + e e e,.43 4 + e + e..44 a a cos + a a cos + a a cos + si >. a a cos + a a ae i ae i ae i + i ae ae i + a e i + a e i + a e i + e i + a cos.. a a cos + a + a cos..45 a cos d, a cos + a a a cos + a cos d a. 6

cos a d a cos + a a.46..3.8 8 8 a <. f log a cos + a.45 a \ a a cos + a + a cos a cos a a cos + a a cos. cos a a cos + a a cos. a. q < q a q a cos q q a q < a a cos. cos a a cos + a a cos a a cos + a a cos + a da a a da. log a cos + a a cos..47 6

...3.9 λ cos log a cos + a d, log a cos + a cos d a. log a cos + a d,.48 log a cos + a cos d a.49 cos m m, ±, ±,.... m. si. 5 {λ } λ >,,... λ cos m m, ±, ±,... λ si. m. δ > [δ, δ]. m m > λ ν cos ν λ ν cos ν m λ ν cos ν. ν ν ν+ si cos ν si ν + si ν si m m λ ν cos ν λ ν si ν + m ν+ ν+ m ν+ λ ν si ν + ν+ λ ν si ν + λ m si m + 63

m ν+ m ν+ λ ν si ν λ ν λ ν+ si ν + λ + si + + λ m si m + λ + si +. [δ, δ] m ν+ λ ν cos ν m ν+ m ν+ λ ν λ ν+ + λ m + λ + si λ ν λ ν+ + λ m + λ + si δ λ + λ m + λ m + λ + si δ λ + si δ. ϵ > λ si δ < ϵ m > > m λ ν cos ν λ + ν+ si δ λ si δ < ϵ. λ cos [δ, δ]. λ cos [m + δ, m + δ]. δ < δ < λ cos λ cos m m, ±, ±,.... λ si m m, ±, ±,.... α >, λ α. 64

6 si α α > cos α m m, ±, ±,.... m..3. 9 9 f log cos < < f +, f f,..47 log a cos + a a cos a <.5. a cos 6 m m, ±, ±,....5 a cos si log cos log si log si cos cos cos < <.,,. t t log si log cos t log cos cos cos t cos 65., < <..5 < t <. < <..5

8 Dii. < < f log cos f+ f,. m m, ±, ±,... fm. φ u f + u + f u f. φ u u du <.53 f <.. φ u u du φ u u du + φ u u < u < < + u < + < { } f + u f + u log cos + u log cos + u. φ u u < { } { } φ u log cos + u + log cos u log cos cos + u cos u log cos log cos u + cos. + cos < u < { φ u log cos + u cos + u cos u log cos { } log. < δ < δ φ u du u lim u + φ u u cos u + cos + cos δ φ u u lim u + du } { } + log cos u log cos du + δ φ u u 66 lim u + φ u u du. si u cos u + cos

δ φ u du. u φ lim u α > u u. u α φ u u α log cos u cos + u u α log u u α log + cos u α log cos u cos + u α u α log u α u α log + cos log si + α u cos u cos lim cos u u u u si, lim t log t. t + α > lim φ u u uα u δ φ u u φ u du. u φ u u du. du. log cos cos < <..54 log log log cos d, cos cos d log cos d,.54 cos cos d log cos d log....55 67

θ log cos θ dθ log..56.56 θ log si d log..57.3. [, ].9 si < <. B. φ ν φ φ + + si ν ν ν ν φ φ < <.58 cos ν ν,,....59 si ν ν +,,,....6 < <.6 cos ν ν ν, si ν ν + + ν ν 3 φ, φ +. ν ν φ. φ φ φ + + φ ν ν si ν ν cos ν ν ν ν ν 3 si ν ν cos ν ν 68

φ + φ.6 φ d φ d, [ ] d cos ν ν d ν φ + d + + ν [ si ν ν + ν ν ] si ν ν [ cos ν ν + d ],.6.6.63 φ. P, P!φ φ d,,....63 P, P P, P d,,... P B [, ] B! ν B + + +!..64.64 3 B! ν cos ν ν,,....64 ν ν si ν ν +,,....65! ζ..66 69

B [, ] B B B..64 [, ] B! ν cos ν ν,..66! ν ν! ζ..67 B!!! ζ ζ 6 4!. B 4!.. B, B, B +,,... B 4!,,,.....68.3. z < B! z zez e z.69..64.65 [, ] B! ν B + + +! cos ν ν,,... ν si ν ν +,,.....65 B!! ζ ζ B z! z ζ. 7

. B! z z < t φ, z B! z.7,, z < B + B + B +! z + + B + B! φ, z + ze z.! z + z z z! φ +, z φ, z + ze z.7. B!. B B z z < t d d φ, z B! zφ, z z B! z z B! z d d φ, ze z d d φ, ze z φ, zze z zφ, ze z φ, zze z φ, ze z gz φ, z gze z..7 gze +z gze z + ze z. gz z e z φ, z zez e z. 7

B B B! z zez e z {B }. B! z z e z B B B B B B B B d B! z zez e z. 7

.3.3 DI log si d log cos d log. 9 log cos cos < <. log cos d log. log cos d log, log si d log. log si d. y y log si O + log si. 73

α >. α log si α log si + α log si + log α log si. lim t log t t + + α log α α log α. lim + α log si. I log si d. t I I I log si t dt. + α log log si d + log si d log si d. log si d t I log cos t dt. I θ log si d + log cos d log si d log si d. I log si d log si θ dθ logsi θ dθ log dθ I log log si d log cos d log si θ dθ I log..7 74

DI cos a d a cos + a a a <,,,,.... 7 a a cos + a + a cos a cos d, a cos + a a a cos + a cos d a. cos a d a cos + a a. a < a cos a a cos. a a a cos + a + a cos..73. 7 a cos + a a cos + a + a cos a., a a cos + a cos, a a cos + a cos a cos + a a cos 4 cos + a 3 a {cos 4 cos + } a, a cos + a cos a cos + a a cos a { cos 4 cos cos + cos } a [ cos {cos + cos } + cos ] 75

a cos + a + a cos a. a < a cos + a a cos + si > a cos + a.73. a a cos + a + a k cos k k cos a cos a cos + a d [ si a. cos d + ] a k cos k cos d k + a cos a d a cos + a a a <,,,.... a a cos + a + a k cos k k a a cos + a d d + a k cos k cos d k [ ] + [ a k si k k. k ] cos a d a cos + a a a <,,,,.... 76

a cos θ + a e iθ ae iθ a eiθ aae iθ e iθ J J e iθ a cos θ + a dθ e iθ e iθ aae iθ e iθ dθ z e iθ, dz izdθ C J C z z aaz z fz dz iz i C z z aaz z z aaz dz C : z e iθ z a Resa lim z a z afz lim z a J i i a a z az a a. cos θ a dθ a cos θ + a a. a a. [, ] [, ] θ ϕ cos θ a cos θ + a dθ cos θ a cos θ + a dθ + cos θ a cos θ + a dθ + cos θ a cos θ + a dθ cos θ a dθ a cos θ + a a cos θ a cos θ + a dθ cos ϕ a cos ϕ + a dϕ. 77

a cos + a e i ae i a C i C dz tz z aaz t <. z a fz tz z aaz Resa lim z a z afz lim z a tz az at a. z e iθ i C dz tz z aaz i ie iθ dθ te iθ e iθ aae iθ dθ te iθ a cos θ + a i dθ te iθ a cos θ + a at a. t t t a cos θ + a { + te iθ + t e iθ + + t e iθ + } dθ a + at + a t + + a t +. e iθ a dθ a cos θ + a a. cos θ a dθ a cos θ + a a..74 cos θ a cos θ + a dθ s θ cos θ a cos θ + a dθ + cos θ a cos θ + a dθ cos s a cos s + a ds cos θ a cos θ + a dθ. 78

.74 cos θ a cos θ + a dθ cos θ a cos θ + a dθ. cos a d a cos + a a a <,,,,....75. a > <.75 a a a. cos a cos + a d a a a >,,,,....76 DI 3 a < log a cos + a d log a a > a f, a log a cos + a, φa log a cos + a d f, a d a < ; a cos + a a cos + si > a cos f, a, f a, a φa. a cos + a φ a a a φa φ. a < f a, a d a cos a cos + a d a cos + a d a a a.75 cos a cos + a d log a cos + a d..77 79

log a cos + a d, { log a cos + a log a } d. a > 3 a φ a 4 4 log a cos + a d log a..78 log cos d log si d log si θ dθ log si d θ log si θ + log dθ 4 log si θ dθ + log 4 log + log. φ 4 4 log + cos d log cos d log cos θ dθ.7 log cos d θ log cos θ + log dθ 4 log cos θ dθ + log 4 log + log..7 DI 4 a a < log a cos + a cos d a a > 8

, a < log a cos + a cos d [ ] log a cos + a si log a cos + a si a si si a cos + a a si si a cos + a d a cos + a cos + a d cos a cos + a d a a + a a a.75 a. log a cos + a cos d a a > a a d d a <,,,.....79 { log a cos + a log a } cos d a log a cos + a cos d a a >,,,.... log a cos + a cos d a. DI 5 log si cos d log cos cos d,,...,,,.....79 a 8

7 {f } δ < δ < a + b [a + δ, b δ] f g g b a f d b a f d b lim a f d [a + δ, b δ] f g. b δ a+δ f d δ +o. b a f d b a b δ a+δ b δ a+δ f d b a [a + δ, b δ] f f d,. b g d < a b b a g d g d a g d < f d b a g d. {f } [a + δ, b δ] f f g f g b a f d b a b a g d. f f. b b f d lim f d f d. a a {f } [a + δ, b δ] f ϵ > > f f < ϵ. b {f f} d a a+δ a a+δ a b δ b {f f} d + {f f} d + a+δ b δ {f f} d b δ b f + f d + f f d + f + f d a+δ b δ 8

a+δ b g d + b a δϵ + g d a b δ b a {f f} d a+δ a b g d + b a δϵ + g d b δ. δ + b a {f f} d b aϵ. b lim a f d b a f d. < a < lim a {a }..79 log a cos + a cos k d ak k..8 f log a cos + a cos k. δ < δ < [δ, δ] f g g a cos + a a cos + si < a < si a cos + si ma { cos + si, cos + si }. g d { f ma log si, ma log 4 si, log 4 } cos. { [ g ma log si, ma log si, log cos ]} f g. log si d, log si d, log cos d δ < δ < [δ, δ] f f log cos 83

. f f log a cos + a log a si log si log a + a si. si + a si [δ, δ] M si δ a a + a si a + M a. a a + M a log a log { a + M a }. ϵ >, > f f log a + a si log a < ϵ, a + M a < ϵ ma { log a, a + M a } < ϵ. 7 lim f d f d. lim a k k log si cos k d. log si cos k d k. θ log cos θ cos kθ dθ k k. 84

3 3. I f I α, β, α + β α, β, y I fα + βy αf + βfy f : I R I I,, < <. f f f f f f 3. y + + + f f + f. B f f f f f. f f f f f f f f f.. f f 3. A, f, P, f, B, f AP AB PB. O A P 85

8 f : I R f f f. f f. I,, < < 3. + f f f f f f f f f. f f f. f f f. < y. F X αfx + βfy fαx + βy X y F X α [f X f αx + βy]. X αx + βy αx + βx αx + βy βx y X αx + βy f X f αx + βy. F X F y F X. F fα + βy αf + βfy. f : I R f f I f. I f f. 3. Γ e t t dt >. ft e t t < < t + ft + g ft dt, g ft dt 86

. < < t α ft e t t α α < α < lim t + tα ft lim t + e t. J. g. g α > g. lim t tα ft lim t +α t e t e t t dt. >. Γ Γ >. g e t t dt lim s + < t < e t t e t t s e t t dt 3. e t t dt <. e t t dt. g. > g >. g e t t e t t e t t dt. e t t dt. < t g. > g >. >. Γ + Γ g + g e t t dt [ e t t ] + e t t dt Γ Γ e t dt [ e t] Γ + Γ, Γ 3.3 87

Γ + Γ Γ Γ!! Γ Γ e t t log tdt e t t log tdt Γ d d e t t dt e t t log tdt.. t t > log t t < t t e t log t dt t e t log t t dt < t e t dt < ϵ. < < lim t + t log t t t log t <.. t t e t log t dt e t log t dt Γ t t γ lim + + + log e t t log t dt e t dt < ϵ e t t log tdt 88.5775664953866...,

R # { R ; \,,,... } 3.4 R # Γ + + 3.5 Γ lim Γ 3.6, R # { R ; \,,,... }. a Γ, I f : I R LC I f >, LC log f I,. > Γ. Γ d log Γ d d d e t t dt, Γ u e t t dt >. Γ Γ Γ Γ {Γ } {Γ } Γ u + Γ u + Γ e t t log tdt, Γ D 4 {Γ } Γ Γ d d log Γ Γ Γ {Γ } b Γ : R # R e t t log t dt e t t [ u + u log t + log t ] dt e t t u + log t dt. {Γ }. Γ e γ lim k e k + k e γ k e k + k 3.7 89

3.7 Γ eγ + +! 3.6 3.8. k + e k. 3.8 k e + + + log + e + e + e e + + + log + e k k c Γ : R # R k Γ Γ γ + k. 3.9 + k k d m d m Γ Γ m+ m! k m+, m. 3. + k 3.7, Γ. I, Γ > 3.7 log Γ γ log + k k { k log + }. k { k log + } k k + k. k k k + k k + k k + k < k k k k log Γ, k + k Γ Γ γ + k + k 9 k,.

. Γ e log Γ,. II R # Γ m, m + Γ m, m Γ., Γ m m, m + Γ. m, m + Γ., m, m u +, u + u, u m, m +. Γ + Γ Γ Γ Γ Γ lim Γ + Γ + u u lim u u u u Γ u u Γ u u u u u {Γ u Γ u } Γ u u u u u. u {u Γ u Γ u } m, m Γ. [ u {Γ u Γ u ] } Γ u u u R # Γ ψ Γ Γ {log Γ }. Γ + Γ Γ + Γ + Γ ψ + Γ + Γ + Γ + Γ Γ + Γ Γ + ψ. R # ψ + ψ +. I, ψ γ + k k + k 9

m m, m + m, m ψ γ + k ψ γ + k k + k k + k. m, m + m, m +. ψ ψ + γ + + k + k + k γ + + k + + k + + k k + k γ + + k + + + k + k k + k γ + + k + + k k k γ + + k + + k + γ + k k k + k III R # Γ R # Γ ii ψ Γ {log Γ } Γ. d m d m ψ m+ m! k m. i m ψ γ + 9 m+, m 3. + k k k + k

+ k. k lim + k + < M M k k k. k k + k k + < M k k ψ,. ψ + ii m k k d m d m ψ m+ m!,. + k + k k k + k + k m+ m+ m +! k + k m+. lim + m+ k + m+ < M k k k M. k m+ k + k m+ k m+ + m+ < M k k ψ m,. d m d m+ ψ m+ m +! k m+ m+,. + k k + k m+ d d log Γ Γ Γ Γ digamma psi ψ d d log Γ 3. 93

3.9 3. ψ γ + k k + k 3.3 d m d m ψ m+ m! k m+, m 3.4 + k 3.3 k k log + γ + O, ψ γ + k γ + k + lim k + k k k lim k γ lim log. + k + k k k k ψ lim log k. 3.5 + k Γ + Γ Γ + Γ + Γ ψ + Γ + Γ + Γ + Γ Γ + Γ Γ + ψ. R # ψ + ψ + 3.6. 3.3 Γ Γ, < < 3.7 si 94

Γ Γ 3.8 Γ eγ k + e k, k Γ e γ k e k. k Γ Γ k k..35 si < < 3.7. 3.7 } {Γ Γ > Γ. 3.4 3.3 f + f, f cosγ 3 G :, R G + G G > G Γ G :, R G + G G G!, G + + + G. G + + G + + + G +... + + + G G + + + G. 95

G +! G G!. < s <,. y log G log G + s log G + s log G log G log G log G log G + log G + log G + s log G + s log G + s log G s G!, G!, G +! log G + log G + s + + s log G + s log G + s log G + log G... log log G + s log G s s G G + s s G. G + s ss + s + Gs s G ss + s + Gs + log, s G ss + s + s G + ss + s + s +. s G + ss + s + Gs s G + ss + s + s + Gs s G + ss + s + Gs. s +. Gs lim s G + ss + s + lim s! ss + s +. Γ :, R Γ + Γ Γ. Γ s lim Gs Γ s < s <. s! ss + s + Gs + ss + s + Gs ss + s + Γ s Γ s +. G Γ! > G Γ., Γ + + 96

Γ lim Γ R # Γ + + + + + + + + + Γ + + Γ + + Γ + 3.8., +, 3.8 3.8. Γ,. Γ m, m + m, m. R # Γ. 3.8 R # Γ. Γ + Γ, R # Γ + f :, R f + f > f fγ G f f. Γ Γ + Γ. 3.9 Γ Γ + Γ + Γ + Γ 3.. 97

9. z z k cos k z k si k 3. + i si k z z z z z z z. eki/ k,,..., z + z + + z + z z z z z z. z z z z z z z z k si k z k cos k i si k cos k + cos k 4 si k si k z z z 3.. k si k k si k. z z z. z k cos k si k si k i si k i si k cos k i cos k + i si k i si k eki/ 98

z z z k i si k eki/ i e i/ f i e i/ i cos + i si m k f 4m+ i 4m cos m + i si m, { f 4m+ i 4m+ cos m + + i si m + } i cos + i si i i, f 4m+3 i 4m+ {cosm + + i sim + } si k.. i cos + i si, { f 4m+4 i 4m+3 cos m + 3 + i si m + 3 i 3 cos 3 + i si 3 i i } f.. z z z i e i/ k si k g Γ Γ G g. + Γ k si k f k si k + Γ + Γ +, G + + Γ + Γ + + Γ + + Γ + Γ + Γ + Γ + Γ + Γ G + + 99

. log G log + log Γ k + k log Γ log G k d d log Γ + k. G G cγ g c Γ. c Γ Γ Γ Γ c Γ. { Γ Γ Γ Γ si si } Γ Γ Γ Γ si Γ Γ 3.7 9. Γ Γ c. g Γ.. c >. log Γ log + log + O

> log g log Γ k k + k { log + + k log + k + k } + O { log + + k log + k } k { + + k log + k + k } log + O k k { log + + k log + k } + + k log + k k log + k log + O + + + k log + k + O. log + + O + k log + k + k { } k + O k + O. + k log + k k k log g log + log + log + log + O k + O + O. + + O log c + log Γ log c + log + log + O log c log + log c.

3.5 33 log! log! log + log + log + O.! Γ > log Γ log + log + µ µ. µ. + log Γ + log + log + log + + µ + log + log + + + µ µ + µ µ + + log +. λ + log + µ µ + λ. µ λ + λ + λ + + λ µ µ + λ. λ λ + log + > λ k k + + k, < λ < + log + t + t, t <. i

t t log + t i ii + log + t t t, t <. ii + {t t }, t <. log + t t k t k+ k +, t <. t + > + log + k k + + k + k k + + k. λ k k + + k. λ > λ < k k k + + k 3 + k 3 +. + < λ < µ 3 + + +. + λ + 3. µ :, R < µ, > 3.3 3

< µ. λ + < + + + m λ + < m m µ. 34 > log Γ log + + + + F log + µ. µ λ +. + m + log + µ µ + λ + + λ + λ µ λ F + + log + + µ + + log + + µ + log + µ + log + + F + log. G e F G + G. λ + log + λ log + + +, λ λ µ f log + + + + + + > λ +. f log, f + > 4

f, F. G 3 G GΓ. log Γ F + C. C log Γ F + C log! log + µ + C. log log! + log + µ + C. 33 < µ log! log + log + log + O log + O µ + C. < µ. lim µ. C log. 34 ψ 35 > < ψ log < log Γ log + log + µ ψ log + µ. < µ < µ λ k > > k k k k + + k+ + k+ + k k+ + + k λ + k + k + + k+ k + + k+ + 3 + +. + [ + ] 5

µ λ + > µ < + + +. 3.6 ψ m B 4m <, B 4m+ > B B! ζ m B 4m m 4m! 4m ζ4m 4m! 4m ζ4m <. m + B 4m+ m 4m +! 4m+ ζ4m + 36 ψ > k ψ ψ 6 3 4m +! 4m+ ζ4m + >. + < ψ < + + 6 3 3.4 + k + k + k + + k + k + k + + k + + k + k + >, k + k + k + + k + k + k + + k + 6 + k 3 + 6 + k + 3 + k + k + 3 + k + 3 + k + 6 + k 3 + k + 3 k 6 + k 3 + k + 3 <. k 6

3.4. 37 >, N + + N k B k k+ < ψ < + N + k B k k+ 3.5 36 N 3.5. 3.4 ψ k + k \,,,... + + N k B k k+ < k + k < + N+ + k B k k+ f f [, + m] f N+ m f + k k +m + N k ftdt + B k k! + B N+ N +! f + m + f f k + m f k m k θ, f f N+ + k + θ +m ftdt +m t dt [ t ] +m + m, f k k!, f N+ N + 3! 7

m k + m + m + + B k k! + m + N ] [ k! + m k+ + k! k+ k + B N+ N +! m N + 3! + k + θ N+4 k m k + m + + + N k B k k+ + m N + m m + N + 3B N+ k k B k + m k+ + k + θ N+4. SN + + N k B k k+, T m, N + m + m N k B k + m k+, m Em, N N + 3B N+ k + k + θ N+4 Em, N B N+ N SN + T m, N < m k < SN + + T m, N +. 3.6 + m > N 3.6 m T m, N T m, N + + + N k B k k+ k + k + N+ + k B k k+ > N + + N k B k k+ < ψ < + N+ + k B k k+ 37 3.5 N, 8

>. + + 6 3 3 5 < ψ < + + 6 3, + + 6 3 3 5 < ψ < + + 6 3 3 5 + 4 7. 3.7 38 >, N log N k N B k < ψ < log k k k k B k k k+ 3.8 f ψ log N B k k k 37 3.5 35 > f ψ N k B k k+ < < ψ log < lim ψ log lim f. > f > log N k B k k k < ψ g ψ log N k g ψ B k k k N k lim ψ log 37 3.5 B k k+ > lim g. > g < ψ < log N k B k k k+ 9

38 3.8 N N N + ψ log N+ { θ. k B 4N+ < ψ 4N+ 4N + log N 38 3.8 k B k < ψ < log N k k k B k k k+ { } log N B 4N+ 4N + k k { } < ψ log N B k k k+ B k k k+ B k k k+ } < θ 4N+ < θ < , f ψ { log k k B k k k+ { } g ψ log B k k k+ } θ, θ. f B 4+ 4 + g B 4 4 θ 4+ < θ < 3.9 θ 4 < θ < 3.3

38 3.8 N, > log < ψ < log + 4, log + 4 6 < ψ < log 5 + 4. 3.3 4 >, N log + log + < log Γ F Γ N k B k kk k < log + N log + k { 38 3.8 log + log + F ψ log + + N lim k N k B k kk k 3.3 B k kk k B k k k+ < [ { Γ log + }] log lim F. > F > log + log + G Γ { 38 3.8 log + N k B k kk k N log + + k G ψ log + N + k lim ψ log B k k k+ > < log Γ } B k kk k }

lim G. > G < log Γ < log + N log + k B k kk k 4 3.3 N N N + log + log + < log Γ N k B k kk k < log + N+ log + k B k kk k < log Γ { log + log + N k B k kk k } < log Γ B 4N+ 4N + 4N + 4N+ { log + log + N k B k kk k } θ. B 4N+ 4N + 4N + 4 3.3 { > log Γ log + θ 4N+ < θ < N log + k B k kk k } > log Γ B 4N 4N 4N 4N { log + N log + k B k kk k } θ. B 4N 4N 4N. θ 4N < θ <

4 >, { F log Γ G log Γ { log + log + log + k log + k B k kk k B k kk k } } F G B 4+ 4 + 4 + B 4 4 4 θ 4+ < θ < 3.33 θ 4 < θ < 3.34 θ, θ. 4 3.33 3.34 lim F, lim G log Γ log + log + k log Γ log + log + k B k kk k B k kk k. log Γ.. log Γ log + log + 4 3.3 N, k B k kk k [ ] 3

> log + < log Γ log + log + 36 3 < log + < log Γ < log + log +, log + 36 3 log + 36 3 + 6 5. 4

4 N.Batir Iequalities for the gamma fuctio[] C.Mortici Ramauja s estimate for the gamma fuctio via mootoicity argumets [] 8 3 + 4 + + 6 < log Γ + < 8 3 + 4 + + 6 3 e 4.. Theorem.6. N.Batir For all positive real umbers we have e + a < Γ + e + b, with the best possible costats a 6.666666 ad b e.765 Theorem.. C.Mortici For every [,, we have + α < Γ + < + β, e e where α 3.33333 ad β 3 39 3.35334. N.Batir e + α < Γ + e + β, with the best possible costats α e.333333 ad β.35 <.35334 3 C.Mortici. N.Batir. N.Batir. + < Γ +. e 3 + < Γ + e 3 log + log + log + < log Γ + log Γ + log 3 5

3.33 log Γ > log + log + 36 3 { log Γ + log log + log + log + } 3 > log + log + 36 3 + log { log + log + log + } 3 log + log + 36 3 log + f 3. f + 4 + 3 63 + 4 + 3 63 + + 3 4 4 + 3 3 + + 3 4 + 3 4 3 4 6 + < f lim f lim log + 3 + log + 36 3 log + log. f >. Γ + + β + β e e, β e. {Γ + } g e, [, g Γ + Γ + e {Γ + } { log + e e } e Γ + Γ + {Γ + } log e {Γ + } {ψ + log } e {Γ + } e ψ log + { } {Γ + } Γ + Γ + log e 6

g < {Γ + } ψ log + < e log Γ + + log ψ log + < log + log G ψ log + < ψ log < < G <. H log Γ + + log G log log + H ψ + + G G log ψ + log + G G G + G G [G] + G G 3.7 > + + 6 3 3 5 < ψ G ψ < 3 6 3 + 3 5. < [G] < 4 [ G ] + G < 3 6 3 + 3 5 6 3 + 3 5 6 3 + 3 5 < 6 3 + 3 5 < 3 5 < 6 3 63 < 3 5 H < g < g [, {Γ + } e, g {Γ + } e g e β [G] + G < > + e < ψ G ψ < + e [G] + G < + e 3 e <. 7

4. NECDET BATIR NECDET BATIR INEQUALITIES FOR THE GAMMA FUNCTION 6 Theorem.6. N.Batir For all positive real umbers we have e + a < Γ + e + b, with the best possible costats a 6.666666 ad b e.765 log ψ > + 4 + 5 6 log + ψ < + 4 5 6 3.3 > log + 4 6 < ψ < log 5 + 4.. g. {Γ + } e, [, g {Γ + } ψ e log + log + ψ <? + 4 5 6 8 3 + 4 + + 6 < log Γ + < 8 3 + 4 + + 6 3 e g < 8 3 + 4 + + 3 3 + 4 5 6. 8

NECDET BATIR INEQUALITIES FOR THE GAMMA FUNCTION.. > F >, F <, F >.. Let be a positive real umber. The the fuctio defied by F log + log log Γ + + log + is strictly completely mootoe i,. strictly completely mootoe 6 + 3 8 A fuctio f is completely mootoic i a iterval I if f has derivatives of all orders i I which alterate i sig, that is f for all I ad,,, 3,... if this iequality is strict for all I ad all o-egative itegers, the f is said to be strictly completely mootoic. > F >, F <, F >. > F log ψ + + + 6 + 3. 8 F + + 3 + 3 3 ψ. 8 ψ k k + k, m + k m + k + m k F + k + + k + + k + k + k k + k + / + k + 3/ k 3 + k + 3/8 3 + k + /8 3 k + k m,, 3 G + k6 + k + k + + k + + k + / k + k + 3/ + k + 3/8 3 + k + /8 3 G t + k > 9

Gt 9799 + 7358t + 6364t + 34446t 3 975 + 89764t4 + 896t 5 + 55594t 6 + 37t 7 975 > F > > F lim log ψ lim F > F < > F > log Γ log + > log + µ log Γ + log + log Γ log + + log + µ. F F log + log log Γ + + log + log + log + log + log log + + log + + log 6 log + µ 6 + 3 µ 8 6 + 3 µ. 8 + 3 8 6 + 3 8 lim µ lim F > F >.

A A. p >, q > Bp, q p q d.. A. J p q d, J J, J. J < p < < α < J. J < q < < α < J. A. t p q d lim + p q + α p lim + α p q lim + q lim p q + α q lim α p q lim p Bp, q A. cos θ t p t q dt Bp, q Bq, p. θ Bp, q cos p θ si q θ dθ. t q t p dt Bq, p. A. A.3 p q B, dθ

B,. A.4 A. + t t Bp, q p t q dt + t + t + t Bp, q t q p+q dt. + t t q p+q dt. A.5 + t A. Γ Γ y e u u du e v v y dv. u u v v y Γ Γ y 4 lim R 4 lim R R R R e u u du R e v v y dv e u +v u v y du dv. R R R S R. R/ S R/ R C R O R R +v u v y du dv < S R/ e u < C R e u S R e u +v u v y du dv +v u v y du dv. 3 R e u u du e v v y dv

lim R S R e u. +v u v y du dv lim R C R e u +v u v y du dv u, v r, θ u r cos θ, v r si θ J u, v r, θ u r v r u θ v θ lim e u +v u v y du dv R C R lim R lim R R R cos θ r si θ si θ r cos θ r e r r cos θ r si θ y rdr dθ e r r +y dr e r r +y dr. r t e r r +y dr e t t +y r cos θ si θ y dθ cos θ si θ y dθ dt r A.3 B, y/... Γ Γ y Γ + yb, y B, y Γ Γ y Γ + y A.6 y A.5 Γ Γ B, e t t +y dt t + t dt Γ + y. A.6 A.7. / si Γ Γ < < A.8 si. [] [] M 3