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Some properties concerning close-to-convexity of certain analytic functions

Abstract

Let f(z) be an analytic function in the open unit disk D normalized with f(0)=0 and f (0)=1. With the help of subordinations, for convex functions f(z) in D, the order of close-to-convexity for f(z) is discussed with some example.

MSC:30C45.

1 Introduction

Let A be the class of functions f(z) of the form

f(z)=z+ n = 2 a n z n

which are analytic in the open unit disk D={zC||z|<1}. A function f(z)A is said to be convex of order α if it satisfies

1+Re z f ( z ) f ( z ) >αin D

for some real α (0α<1). This family of functions was introduced by Robertson [1] and we denote it by K(α).

A function f(z)A is called starlike of order α in D if it satisfies

Re z f ( z ) f ( z ) >αin D

for some real α (0α<1).

This class was also introduced by Robertson [1] and we denote it by S (α). By the definitions for the classes K(α) and S (α), we know that f(z)K(α) if and only if z f (z) S (α).

Marx [2] and Strohhäcker [3] showed that f(z)K(0) implies f(z) S ( 1 2 ).

This estimate is sharp for an extremal function

f(z)= z 1 z .

Jack [4] posed a more general problem: What is the largest number β=β(α) so that

K(α) S ( β ( α ) ) .

MacGregor [5] determined the exact value of β(α) for each α (0α<1) as the infimum over the disc D of the real part of a specific analytic function. It has been conjectured that this infimum is attained on the boundary of D at z=1.

Wilken and Feng [6] asserted MacGregor’s conjecture: If 0α<1 and f(z)K(α), then f(z) S (β(α)), where

β(α)={ 1 2 α 2 2 ( 1 α ) 2 if  α 1 2 , 1 2 log 2 if  α = 1 2 .
(1)

Ozaki [7] and Kaplan [8] investigated the following functions: If f(z)A satisfies

Re f ( z ) g ( z ) >0in D

for some convex function g(z), then f(z) is univalent in D. In view of Kaplan [8], we say that f(z) satisfying the above inequality is close-to-convex in D.

It is well known that the above definition concerning close-to-convex functions is equivalent to the following condition:

Re z f ( z ) g ( z ) >0in D

for some starlike function g(z)A.

Let us define a function f(z)A which satisfies

Re z f ( z ) g ( z ) >αin D

for some real α (0α<1) and for some starlike function g(z) in D.

Then we call f(z) close-to-convex of order α in D with respect to g(z).

It is the purpose of the present paper to investigate the order of close-to-convexity of the functions which satisfy f(z)K(α) and 0α<1.

2 Preliminary

To discuss our problems, we have to give here the following lemmas.

Lemma 1 Let p(z)=1+ n = 1 c n z n be analytic in D and suppose that

p(z) 1 α z 1 + β z in D,

where means the subordination, 0<α<1 and 0<β<1.

Then we have

1 α 1 + β <Rep(z)< 1 + α 1 β .

This shows that

Rep(z)>0 in D.

A proof is very easily obtained.

Lemma 2 Let p(z)=1+ n = 1 c n z n be analytic in D, and suppose that there exists a point z 0 D such that

Rep(z)>c for |z|<| z 0 |

and

Rep( z 0 )=c,p( z 0 )c

for some real c (0<c<1). Then we have

Re z 0 p ( z 0 ) p ( z 0 ) { 1 c 2 c when 1 2 c < 1 , c 2 ( 1 c ) when 0 < c < 1 2 .

Proof Let us put

q(z)= p ( z ) c 1 c ,q(0)=1.

Then q(z) is analytic in D and

Req(z)>0for |z|<| z 0 |

and

Req( z 0 )=0,q( z 0 )0.

Then, from [[9], Theorem 1], we have

z 0 q ( z 0 ) q ( z 0 ) =i,

where

1 2 ( a + 1 a ) 1when argq( z 0 )= π 2

and

1 2 ( a + 1 a ) 1whenargq( z 0 )= π 2 ,

where q( z 0 )=±ia and a>0.

For the case argq( z 0 )= π 2 , we have

z 0 q ( z 0 ) q ( z 0 ) = z 0 p ( z 0 ) p ( z 0 ) c =i

and so

If we put

h(x)= 1 + x 2 c 2 + ( 1 c ) 2 x 2 (x>0),

then it is easy to see that

1 c 2 <h(x)< 1 ( 1 c ) 2 when  1 2 c<1

and

1 ( 1 c ) 2 <h(x)< 1 c 2 when 0<c< 1 2 .

This shows that

Re z 0 p ( z 0 ) p ( z 0 ) { 1 c 2 c when  1 2 c < 1 , c 2 ( 1 c ) when  0 < c < 1 2 .

For the case argq( z 0 )= π 2 , applying the same method as above, we have the same conclusion

Re z 0 p ( z 0 ) p ( z 0 ) { 1 c 2 c when  1 2 c < 1 , c 2 ( 1 c ) when  0 < c < 1 2 .

This completes the proof of the lemma. □

Our next lemma is

Lemma 3 Let p(z)=1+ n = 1 c n z n be analytic in D and suppose that there exists a point z 0 D such that

Rep(z)>c for |z|<| z 0 |

and

Rep( z 0 )=c,p( z 0 )c

for some real c (c<0).

Then we have

Re z 0 p ( z 0 ) p ( z 0 ) > c 2 ( 1 c ) >0.
(2)

Proof Let us put

q(z)= p ( z ) c 1 c ,q(0)=1.

Then q(z) is analytic in D. If p(z) satisfies the hypothesis of the lemma, then there exists a point z 0 D such that

Req(z)>0for |z|<| z 0 |

and

Req( z 0 )=0andq( z 0 )0,

then p(z) satisfies the conditions of the lemma.

For the case argq( z 0 )= π 2 , applying the same method as in the proof of Lemma 2, we have

Re z 0 p ( z 0 ) p ( z 0 ) = ( 1 c ) c a c 2 + ( 1 c ) 2 a 2 c ( 1 c ) 2 ( 1 + a 2 c 2 + ( 1 c ) 2 a 2 ) .

Putting

h(x)= 1 + x 2 c 2 + ( 1 c ) 2 x 2 (x>0),

it follows that

h (x)= ( 2 c 1 ) x ( c 2 + ( 1 c ) 2 x 2 ) 2 <0(x>0).
(3)

Therefore, from (3) we obtain (2) .

For the case argq( z 0 )= π 2 , applying the same method as above, we have the same conclusion as in the case argq( z 0 )= π 2 . □

3 The order of close-to-convexity

Now, we discuss the close-to-convexity of f(z) with the help of lemmas.

Theorem 1 Let f(z)A, and suppose that there exists a starlike function g(z) such that

  1. (i)

    for the case 1 2 c<1,

and

  1. (ii)

    for the case 0<c< 1 2 ,

Then we have

Re z f ( z ) g ( z ) >c in D.

This means that f(z) is close-to-convex of order c in D.

Proof Let us put

p(z)= z f ( z ) g ( z ) ,p(0)=1.

Then it follows that

1+ z f ( z ) f ( z ) = z g ( z ) g ( z ) + z p ( z ) p ( z ) .
  1. (i)

    For the case 1 2 c<1, if there exists a point z 0 D such that

    Rep(z)>cfor |z|<| z 0 |

and

Rep( z 0 )=c,

then, applying Lemma 2 and the hypothesis of Theorem 1, we have

p( z 0 )c

and

Re z 0 p ( z 0 ) p ( z 0 ) 1 c 2 c .

Thus, it follows that

1 + Re z 0 f ( z 0 ) f ( z 0 ) = Re z 0 g ( z 0 ) g ( z 0 ) + Re z 0 p ( z 0 ) p ( z 0 ) Re z 0 g ( z 0 ) g ( z 0 ) 1 c 2 c ,

which contradicts the hypothesis of Theorem 1. (ii) For the case 0<c< 1 2 , applying the same method as above, we also have that

Re z f ( z ) g ( z ) >cin D.

This completes the proof of the theorem. □

Applying Theorem 1, we have the following corollary.

Corollary 1 Let f(z)A be convex of order α (0<α<1), and suppose that there exists a starlike function g(z) such that

  1. (i)

    for the case 1 2 α<c,

and

  1. (ii)

    for the case 0<α<c 1 2 ,

Then we have

Re z f ( z ) g ( z ) >β(c)>β(α)>α in D.

Remark 1 For the case 0<α<c<1, it is trivial that

α<β(α)<β(c)<1.

Example 1 Let f(z)A satisfy

1+Re z f ( z ) f ( z ) >Re 1 A z 1 + A z 1 β ( 1 2 ) 2 β ( 1 2 ) > 1 10 in D,
(4)

where

A= 32 β ( 1 2 ) 10 8 β ( 1 2 ) + 10 0.29605

and β( 1 2 )= 1 2 log 2 . If we consider the starlike function g(z) given by

g(z)= z ( 1 + A z ) 2 ,

then we have

Re z f ( z ) g ( z ) >β ( 1 2 ) 0.7213,

which means that f(z) is close-to-convex of order β( 1 2 ) in D.

Next we show

Theorem 2 Let f(z)A and g(z)A be given by

g(z)={ z ( 1 + β z ) α + β β ( β 0 ) , z e α z ( β = 0 )

for some α (0α<1) and some β (0β<1). Further suppose that for arbitrary r (0<r<1),

min | z | = r ( Re z f ( z ) g ( z ) ) = ( Re z 0 f ( z 0 ) g ( z 0 ) ) | z 0 | = r z 0 f ( z 0 ) g ( z 0 )

and

1+Re z f ( z ) f ( z ) c 2 ( 1 c ) + 1 α 1 + β

for c<0. Then we have

Re z f ( z ) g ( z ) >c in D.

Proof Let us define the function p(z) by

p(z)= z f ( z ) g ( z ) ,p(0)=1

for c<0. If there exists a point z 0 D such that

Rep(z)>cfor |z|<| z 0 |

and

Rep( z 0 )=c

for c<0, then from the hypothesis of Theorem 2, we have

Rep( z 0 )p( z 0 ).

Therefore, applying Lemma 1 and Lemma 3, we have

1 + Re z 0 f ( z 0 ) f ( z 0 ) = Re z 0 p ( z 0 ) p ( z 0 ) + Re z 0 g ( z 0 ) g ( z 0 ) > c 2 ( 1 c ) + 1 α 1 + β .

This is a contradiction, and therefore we have

Re z f ( z ) g ( z ) >cin D.

 □

Remark 2 In view of the definition for close-to-convex functions, if f(z) satisfies

Re z f ( z ) g ( z ) >0in D,

then we can say that f(z) is close-to-convex in D. But c should be a negative real number in Theorem 2. Therefore, we cannot say that f(z) is close-to-convex in D in Theorem 2.

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The authors thank the referees for their helpful comments and suggestions to improve our manuscript.

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Correspondence to Melike Aydog̃an.

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All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.

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Nunokawa, M., Aydog̃an, M., Kuroki, K. et al. Some properties concerning close-to-convexity of certain analytic functions. J Inequal Appl 2012, 245 (2012). https://doi.org/10.1186/1029-242X-2012-245

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Keywords

  • analytic
  • starlike
  • convex
  • close-to-convex
  • subordination