Some Properties Concerning Close-to-Convexity of Certain Analytic Functions

Scanned by CamScanner

] {d •

n

T h i s e s t i m a t e i s s h a r p f o r a n c x t r e m a l f u n c t i . n

M a r x [ 2 1 a n d S t r . t , h / . k . . 13 1 s l : " " d t h a t l ( : ) A ( 0 ) i m p [ i - ! ( : ) ‹ s-

{ : ) .

t i o n s f o , t h e c l a s s e s 人 ( . ) a n d s-

(a ) . W e k n u l . T h a t. f ( : ) ‹ K ( a ) , f a n d 0 • h :

i t f ( z l e s-

(. ) .

T h is c l a s s m s a l s . I n t r o d u c . D b y R . B c n " " [ 1 ] • . d . E d e n o t e i t l , y s

’

(. ) B v t h . d . I i

fo r s o m e r e a l a ( 0 . . I ) -

A 1" " i o • f ( z ) ‹ ¯ i s c a l l e d s t a r l i k e o f . Rd . . a i n 3 i l i t . A t i . l i . .

. e d e n o t c i t b y A’

(. )

fo r s o m e r e a l . ( 0 / . . 1 ) . T h is f a m i l y o f 1. N c t i o n s . A s i n t . . d . c . d b v R o b e r t s o n ï 1 ! a n d

bc c o n . E x o f o r d e r ° i f i t m t i . Ri . .

. h i c h a r e ’ " I p i c i n t h e . p e n u n i t d i s k 3 - (= ‹ : L. . I 1 A f u n c l l o n f ( z ) ‹ A I . M i d t o

1. . t . 4 bc t h e c l a s s o f 1. . I . ". F[ z l o f t h e f o r m

1 I n t r o d u c t i o n

Kw w o r d s : a n a l& ; , t a " , k t : ; c o n v e c l o s e •

t o •

c o n v e x . S u bo , d , n a u o n

M SC : n i m a ’ ! 3 0 C 4 5

- ’ " ¢ u a m c . . c c l o se t O C a n \ T x l t y f o : í u ) 1s d r sc u s s e d w , t h l o r n e e m ’ " e

- - 1 34 9 60 ; r. w r ( 0 ) = 1 . W i : h t h e h e lp o f Su b o , d , n a t i o n s . Fo , c o n v e x f u n c t i o n s f ) n D . T h e o r u e , o f

m - v ’ " w . S lc L e : í ) b e a n a n a l

y T : c f " " 10 . i n t h e o F e n u n i t d i s k D . O r m a l , z e d w n h f r o ) = 0 a n d

em u - m - u A b s t r a c t

’

c- e

M a m o ru N u n ß k ; > . •

. 3 M e like : l d o ! a n"

. Kaz u o K u r o k i.

I s m e t Yi ld , ï :

a n d S h : g

ey

o s h i O w a

’

f u n c t i o n s

c l o s e - t o - c o n v e x i t y o f c e r t a i n a n a l y t i c

So m e p r o p e r t i e s c o n c e r n i n g

E A R C O p e n A

’ " - IOW ~ b D r 1e q - 14 - ’ " " u . K c o m l c a n t e n u 2 0 \ u u 2 4 5

0 ?o u r n a l o f In e q u a l i t i e s a n d A p p l i c a t i o n s

"w


TT¸° R e p ( z ) • I •

1 -

( r 1 + a

T ¢Be n w b a v c

w h e 1z « m e a n s t ¢Be s u h o r d i n a t i o n , O a « \ a . R 1 t i O ç ¢l « l .

p (z ) <

i + ¢ z m ,

1 -a z

L e m m a 1 L t t p \ z \ • \ + : . \c

. . Be a n a l y ¢i c i n a n d s u p p o se t h a t

16 d is c u s s o u r p r o b l e m s , w e h a v e t o g

i v e h e r e U t c f o l l o w i ng

l c n t n 1 a s .

2 Pr e l i m i n a r y

t h e f u n c 1i o n s w h i c h s a l i s f y j

’

(£ ) ‹ L (a ) a n d 0 ç u < 1 .

I t i s t h c p u r p o s e o f t h e p r e s e n t p a p e r t o i n v e s t i g a t e t h c u r d c r o f c l o s e -

t o-

c o n v e x i t y o t

T h e n w e t a l l i ( z ) c l a w -t o

-

c o n v e x o f o r d e r a i n w i t h r e s p e c t t o g t z ) .

f o r s o m e r e a l a ( 0 : ; . . 1 ) a n d f o r s o 1n e s t a r l i k c t u n e l : i o n g ( z ) 1n .

g ( z )

Rc > « i n =

L e t u s d e l i n e a l u n c t i o n ! ( z ) ‹ A w h i c h s a t i s f i e s

f o r s o m e s t a r l i k e f u n c t i o n g ( : ) ‹ . 4 .

R c í . 0 i n

a i e n t t o t h c f i ) I 1o w i n g c o n d i t i o n :

I t is w e l l k n o w n t h a t t h e a b o v e d e f i n i t i o n c o n c c r n i n g c l o s e • t o

• c o n v e x f u n c t io n s i s e q u i . -

t h a t í ( z ) s a t i s f y i n g t h e a b o v e i n c q u . 1i t y i s c l o s e -

t o-

c o n v e x i n = .

fu r s o m e c o n v e x f u n c t i o n g ( z ) , t h c r 1 j’

(t ) I . U n i . A l . N l i n ï : . I n 1 : i e w . 1 K a p l a n [ 8 1 , w c s ay

R c > o i n î )

O z a k i B : : 1 a n d K a p l a n 18 ] i n v e s t i g a t e d t h e f o l l o w i n

g í u n c t i n n s : I t j

’

(z ) ‹ A s a t i s f i e s

11)

w ’ i l k c n a n d F e n g [ ó 】 a s s e r t e d M . C G , c g " : c o . j . I u r e : l f o / . . 1 a n d f ( , ) ‹ L ( . ) . T h e n

t h is i n f i m u m i s a t t a i n e d o n t h e b o u n d a • : : o f = a t = - -1 .

o v e r t h e d i s c n . F t h e r e a l p a r t o f a s p e c i f i c a n a l y t i c f u n c t i o n . 1 t h a s b e e n c n n ie c t u • . d t h a t

M " G " g • • 15 ] d c t c m l i n e d t h e e x a c t v a l u e o f H ( - ) f o > e a c h . ( 0 • • I ) a s t h e I . I ì rn u n \

1. . k 14 j p . . E d a n l o r c g c ’ " . 1 p . . B l " " W i 1a t i s t h e I . "g

’ " u m b c r JS • p [ . ) s o t h a t

h ’ " "’

: - ’" " " l d i . q . . 1 1u ’ " d ’

" l • . t " " " " c - t " " 0 1 2 2 4 5

N . No * - . C l a l h - . 1 . F h. Eq u o f 1ï , e s ’ " A p p l i t M , O n , 2 0 1 2 . 2 0 1 2 1 4 5 p .

g. 2 9


. h e r e q ( : . ) = 士 a n d n . 0 .

a n d

w h e r e

T h e n , f r o m 19 , H \ e o r e m I I , w e h a v e

Rc q ( 30 ) = 0

q ( Z o ) 1 0 .

a n d

R c q ( z ) > 0 f o r 1z l < Iz . .

T h e n q ( r ) i s . n . I y t i c i n D a n d

P r u q f L e t u s p u t

R c p ( : , : ) = c , p { z . ) H

: 0 ‹ s 1Bd , t l 1a l

L e m m a 2 L N H : ) = \ • : , c . . e’

b e a Bm B ! 1 [ c i n n . a H 1 t s np p

o s c t i l a t t B 1 c r e e x i s • I

• I "

A p ’ " F i ’ " r y . . s i [ o b t a i n e d •

Rc pt ) > 0 i n D .

T h i s s l , 0 w s t B, a t

Pa g • 3 . R9

h " : / / -J

• " " r ’ " • q

. R 1 u " " d ’" l

’ " " . . . / c . t . T l 2 0 1 0 1 / 2 4 5

N . . h . . A d . I 0 . . 1 o r b t e r r. M " " d d

ww ° " " 0 1 2 . 2 0 1 2 9 4 5

r a g e 1 u 1


z o k sh e l l t i n t

m m a 3 1̆ t l 1\ z ) = \ + : , c . . :

•

f nanal i c in m 1d s t 1p p o : i c t J 1a t t l 1e r e e x s s a pn i w

O u r n e x t l e m m a i 5

丁 h i ?i c o m pï e t c • t h e p

r o o f . E t h = l e m m a . 0

s i o n

Fo r t h c c a s e a r g q [ o ). - ] , a

p pl

yi n

g t h = s a m e m e t h o d a s a b o w , w e h a v e t h = s a n 1 c c o n c

l

T h i s s h o w s t h a t

a n d

t h e n i t i s e a s : : t o s e e t h a t

1f W C p u t

a n d SO

Fo r t h e c a s e " g q f . . . 1 = ] . W C h a v e

p . Ge 4 ’ "

h t cp " - j " " . 1. % . . q . " , t ’ " ’ " P p l ’ " " " " " " " 0 2 0 1 D I / 2 4 5N . . - u d h . . 1d l i n e q . ’ i l . . N d Jw n . Fi o n s 2 0 1 2 , 2 0 1 2 . 2 4 5

r a g c


D

T h e o w m \ L e ¢ f t z ) A , a n d 5

B 1p p

o s e t i l a t t h e r e e x i s t s a s m r Bi k r ì i1ï l c l i o l 1 g ( 1 l s1 1 c h l i m t

N o w , w e d i s c u s s t l w c l o s e -t o

-c o n v e x i t y o f f (z ) w i t h t h c h e l p o í l c 1n 1n a s .

3 T h e o r d e r o f c l o s e - t o -c o n v e x it y

c l u s i o n a s i n t h e c a s e a r g q ( z o ) • ì

Fo r t h e c a s e a 1g i 1 ( z o ) . - å , a pp ly i n 吕 t h e s a m e m e t h o d a h a b n v = , w e ha v e t h c s a m e c o

T h e r e f o r e , f r o m ( 3 1 w e o b t a i n ( 2 ) ,

AB)h’

(x ) . ( h < 0 ( x » 0 ) .

i t fo l l o w s t h a t

h (x ) •

L J

. (1 -

c ) ’x

J ( x > t ) ) ,

Pu t t i n g

ha v e

Fo r t h c c a s e u r g q ( Z u ) • , ap p

l. •

i ng

t h c sa m e m e t h o d a s i n t h c p r o o i o l-

1. R 1 n m n 2 , \ e e

t h e n 11 ( z ) s a t i s 1i e s t h c c o n d i t i o n s n l t h c l e m m a .

R c q ( : u ) = 0 a n d q (: , l ) n ,

a n d

p o i n t Z o c D s u c h t h a i

T h e n q ( z ) i s a n a l v t i c i n D . I l p t : ) s J t i : ; r i c s t h e h y p, \ t h c ’ i s n t t h e l > 1n n m , l h r r i 1 h r n

’c x i s t » I \

1’ ’ " Y 1 . R l u s p n t

I’

l i c H H V Bu B> e

/b • s o n t e r «1 l 1 e ( , • : 0 ) .

n r n 1

h t 1p j / w w . l

• h

q. . i

l1 l , d a

11

pl k , t l l o n i . c o m / c c h I t e , , t / 1 o i 1 . l / 1 4 5

N . . o l i n . , , e 1 M h " " o 1 i : . Q m , h n J p 1 n r M JW

l k . N w . . 2 0 1 ? . 2 0 1 1 1 4 1


a n d

p (z o ) c

th e n , appl ! ng Le m m a 2 a n d t h e h y p o t h e s i s o l T h e o r e m 1 . W e ha v e

Rc p ( z o ) - c .

a n d

Rc p ( e ) > ¢ Fo r l z l < l z o l

(n Fo r t 11c c a s e . S c « l . I f t h e r e e x i s t s a p

o i 1 u z o ‹ 3’

s u c h t h a t

T h e n i t ó u o w i t h a t

P r o o f L c t u s p u t

TJˇ I s Bn e a B1s r J1a t f t : \ i s c l a s e • t o

-c o i n

’e x o f . . D c r c i n D .

T i n n . t BBa \ w

( i i ) A1 • r h c c n M 0 . . . # •

n n d

I . R c . R c _I 1 1 D .

P° F • 6 ’ "

h H p J - . . 2 0 1 2 0 2 4 5

N . - . E t d Jo . . ¢o r rm q - h , I . . N d A p p l ’ ’ " ’ " 0 1 2 . 2 0 1 2 2 4 5


bl i d ) + 10A =

3 2 9 ( ) - 10. 0 . 296 0 5

w h e r e

(4 )

Ex a m p l e 1 L e t l ( z ) ‹ A s a t i s f y

Re m a r k 1 F o r t h e c a s e 0 ( a < c < 1 , i t i s t r i v i a l t h a t

Re Ý 3 > / I ( c ) > ß { a ) > a i n D ,

11æ H w e h d v e

ダ芦回向彐

(i i ) þ r t h e m s c 0 < a . c : 1 ’

, 孑H イ

プ卢(C ) 加口

(i w r t h e m s e ì # a • c ,

M « r 1i k e f u n c t i m 1 g \ z \ s m e l t t h a t

C o r o l 1a r y l L e t f (妁‹ AbecQnwxoí or de" ( 0. a<\ ) , a r r d s B tp p

o s e t i l a t t 1 ï e r e e . . I s t s a

A p p ll : i n g T h c . Re m 1 , w e h a v e t h e f o l l " " i ng

c o r o l l a r y •

T h i s c o m p l e t e s t h e p r o o f o f t h e t h e o r e n r .

g ( z )

J

R c . c i n ; : .

s a m e " " h o d a s a b . . c . . A l s o h a v e t h a t

w h i c h c o n t r a d i c t s t h e h y p o t h e s i s o l T \ 1e o r c n \ 1 . ( i i ) Fo r t h e Ca Se 0 . = • / . Ap p

ll

: in g t h e

< R . m’ -

c

T h u s , i t f o l i o \ . s t h a t

Pa g . 7 o f q


Rc p ( z 11) p ( e n ) .

fo r e < 0 , t he n f r o m t i l e h y p o t h e s i s o f T h c o r c m 2 . w e ha v e

Rc p ( z o ) = c

a n d

Rc p ( z ) > c f o e BB l < l z t , I

fo r e ( 0 . i r t h e r e e x i s t s a p o i n t Z o ‹ u • s u c h th a t

p (z ) = ˇb, p (0 ) = 1

P r o q í L e t u s d e l i n o t h c f u n c l : i o n p ( z ) b !’

R c ï ¥ > r i n D .

JÒ • c ( 0 . T I1E l l w g ï 1a w e

a n d

þ r s i n e a ( O : ; a « l ) m r1 s o 1n r p ( 0 3 i $ . n - Fm t h o r s H p p o sc t » ° j ° ’ " b i t ’ " y r

T h e o r e m 2 L e i f ( z ) ‹ A m n 1 g ( z ) ‹ A b c g ’ " m b y

N e x t W c s h o w

w h i c h m e a n s t h a t í t z ) I . c l o s e -t o

• c o n . C x o f o r d e r ß d ) i n .

t h e n . e h a w

g ( z ) =

-

a n d J4 C] ) • : I ; : n - 1 F U T c o n s i d e r t h e s t a r l i k e f u n c t i o n g t . } g i ’ " b )

’

h 11p j / w w w j l . . q _ t i l l , p p l • . . , . m / u 2 0 1 u l / 2 4 s

N . . K a . . E r a r . I 0 " " o H n o1

" " n e s ’ " d, wl k ’ " " 0 1 2 . 2 0 1 2 9 4 S P a

ge a F

P a g e 1 o f 1

Some Properties Concerning Close-to-Convexity of Certain Analytic Functions

Documents

Transcript of Some Properties Concerning Close-to-Convexity of Certain Analytic Functions