# On logarithmic coefficients of some close-to-convex functions

###### Abstract.

The logarithmic coefficients of an analytic and univalent function in the unit disk with the normalization is defined by . Recently, D.K. Thomas [On the logarithmic coefficients of close to convex functions, Proc. Amer. Math. Soc. 144 (2016), 1681–1687] proved that for functions in a subclass of close-to-convex functions (with argument ) and claimed that the estimate is sharp by providing a form of a extremal function. In the present paper, we pointed out that such extremal functions do not exist and the estimate is not sharp by providing a much more improved bound for the whole class of close-to-convex functions (with argument ). We also determine a sharp upper bound of for close-to-convex functions (with argument ) with respect to the Koebe function.

###### Key words and phrases:

Univalent, starlike, convex, close-to-convex, logarithmic coefficient.###### 2010 Mathematics Subject Classification:

Primary 30C45, 30C55## 1. Introduction

Let denote the class of analytic functions in the unit disk normalized by . If then has the following representation

(1.1) |

We will simply write when there is no confusion. Let denote the class of all univalent (i.e. one-to-one) functions in . A function is called starlike (convex respectively) if is starlike with respect to the origin (convex respectively). Let and denote the class of starlike and convex functions in respectively. It is well-known that a function is in if and only if for . Similarly, a function is in if and only if for . From the above it is easy to see that if and only if . Given and , a function is said to be close-to-convex with argument and with respect to if

(1.2) |

Let denote the class of all such functions. Let

be the classes of functions called close-to-convex functions with respect to and close-to-convex functions with argument , respectively. The class

is the class of all close-to-convex functions. It is well-known that every close-to-convex function is univalent in (see [2]). Geometrically, means that the complement of the image-domain is the union of non-intersecting half-lines.

The logarithmic coefficients of are defined by

(1.3) |

where are known as the logarithmic coefficients. The logarithmic coefficients play a central role in the theory of univalent functions. Very few exact upper bounds for seem have been established. The significance of this problem in the context of Bieberbach conjecture was pointed out by Milin in his conjecture. Milin conjectured that for and ,

which led De Branges, by proving this conjecture, to the proof of the Bieberbach conjecture [1]. More attention has been given to the results of an average sense (see [2, 3]) than the exact upper bounds for . For the Koebe function , the logarithmic coefficients are . Since the Koebe function plays the role of extremal function for most of the extremal problems in the class , it is expected that holds for functions in . But this is not true in general, even in order of magnitude [2, Theorem 8.4]. Indeed, there exists a bounded function in the class with logarithmic coefficients (see [2, Theorem 8.4]).

By differentiating (1.3) and equating coefficients we obtain

(1.4) |

(1.5) |

(1.6) |

If then follows at once from (1.4). Using Fekete-Szegö inequality [2, Theorem 3.8] in (1.5), we can obtain the sharp estimate

For , the problem seems much harder, and no significant upper bound for when appear to be known.

If then it is not very difficult to prove that for and equality holds for the Koebe function . The inequality for extends to the class was claimed in a paper of Elhosh [4]. However, Girela [6] pointed out some error in the proof of Elhosh [4] and, hence, the result is not substantiated. Indeed, Girela proved that for each , there exists a function such that . In the same paper it has been shown that holds for whenever belongs to the set of extreme points of the closed convex hull of the class . Recently, Thomas [12] proved that for functions in (close-to-convex functions with argument ) with the additional assumption that the second coefficient of the corresponding starlike function is real. Thomas claimed that this estimate is sharp and has given a form of the extremal function. But after rigorous reading of the paper [12], we observed that such functions do not belong to the class (more details will be given in Section 2).

By fixing a starlike function in the class , the inequality (1.2) assertions a specific subclass of close-to-convex functions. One of such important subclass is the class of close-to-convex functions with respect to the Koebe function . In this case, the inequality (1.2) becomes

(1.7) |

and defines the subclass . Several authors have been extensively studied the class of functions that satisfies the condition (1.7) (see [5, 7, 9, 11]). Geometrically (1.7) says that the function has the boundary normalization

and is a domain such that for every . Clearly, the image domain is convex in the positive direction of the real axis. Denote by the class of close-to-convex functions with argument and with respect to Koebe function . That is

Then clearly functions in are convex in the positive direction of the real axis. In the present article, we determine the upper bound of for functions in and .

## 2. Main Results

Let denote the class of analytic functions with positive real part on which has the form

(2.1) |

Functions in are sometimes called Carathéodory function. To prove our main results, we need some preliminary lemmas. The first one is known as Carathéodory’s lemma (see [2, p. 41] for example) and the second one is due to Libera and Złotkiewicz [10].

###### Lemma 2.1.

In [12], Thomas claimed that his result (i.e. ) is sharp for functions in the class by ascertaining the equality holds for a function defined by where with and with , . But in view of Lemma 2.2, it is easy to see that there does not exist a function with the property , . Thus we can conclude that the result obtained by Thomas is not sharp. The main aim of the present paper is to obtain a better upper bound for for functions in the class than that of obtained by Thomas [12]. To prove our main results we also need the following Fekete-Szegö inequality for functions in the class .

###### Lemma 2.3.

For (close-to-convex functions with argument ), we obtained the following improved result for (compare [12]).

###### Theorem 2.1.

If then .

###### Proof.

Let be of the form (1.1). Then there exists a starlike function and a Carathéodory function of the form (2.1) such that

(2.2) |

A comparison of the coefficients on the both sides of (2.2) yields

By substituting the above and in (1.6) and then further simplification gives

(2.3) | ||||

In view of Lemma 2.2 and writing and in terms of we obtain

(2.4) | ||||

where and . Note that if denote the third logarithmic coefficient of then

(2.5) |

Since the class is invariant under rotation, without loss of generality we can assume that , where . Taking modulus on both the sides of (2.4) and then applying triangle inequality and further using the inequality (2.5) and Lemma 2.3, it follows that

where we have also used the fact . Let where and . For simplicity, by writing we obtain

(2.6) |

where and

Thus we need to find the maximum value of over the rectangular cube .

By elementary calculus one can verify the followings:

We first find the maximum value of on the boundary of , i.e on the six faces of the rectangular cube .

On the face , we have , where . Thus

On the face , we have , where .

On the face , we have , where . By using elementary calculus it is easy to see that

On the face , we have , where . We first prove that in the interior of . On the contrary, if in the interior of then

and hence

(2.7) |

Further, (2.7) reduces to

which is equivalent to and . This contradicts the range of . Thus in the interior of .

Next, we prove that has no maximum at any interior point of . Suppose that has the maximum at an interior point of . Then at such point and . From , (for points in the interior of ), a straight forward calculation gives

(2.8) |

Substituting the value of as given in (2.8) in the relation and further simplification gives

(2.9) |

It is easy to show that the function is strictly increasing in . Since and , the equation (2.9) has exactly one solution in . By solving the equation (2.9) numerically, we obtain the approximate root in as . But the corresponding value of obtained by (2.8) is which does not belong to . Thus has no maximum at any interior point of .

Thus we find the maximum value of on the boundary of . Clearly, ,

and

By using elementary calculus we find that

Hence,

On the face ,

where and . Differentiating partially with respect to and and a routine calculation shows that

where . Now we find the maximum value of on the boundary of and on the set . Note that

On the other hand by using elementary calculus, as before, we find that

where denotes the boundary of . Hence, by combining the above cases we obtain

On the face ,

where and . Differentiating partially with respect to and and a routine calculation shows that

where . Now, we find the maximum value of on the boundary of and on the set . By noting that

and proceeding similarly as in the previous case, we find that

Let . Then

We prove that has no maximum value at any interior point of . Suppose that has a maximum value at an interior point of . Then at such point , and . Note that , and may not exist at points in . In view of (for points in the interior of ), a straight forward but laborious calculation gives

(2.10) |

Substituting the value of as given in (2.10) in the relations and and simplifying (again, a long and laborious calculation), we obtain

(2.11) |

and

(2.12) |

Since , solving the equation (2.12) for , we obtain

(2.13) |

Substituting the value of in (2.11) and then further simplification gives

Taking the last term on the right hand side and squaring on both sides yields

(2.14) |

Clearly in . On the other hand the polynomial has exactly two roots in , one lies in and another lies in . This can be seen using the well-known Strum theorem for isolating real roots and hence for the sake of brevity we omit the details. By solving the equation numerically, we obtain two approximate roots and in . But the corresponding value of obtained from (2.13) and (2.10) are and which do not belong to . This proves that has no maximum in the interior of

∎

We obtained the following sharp upper bound for for functions in the class .

###### Theorem 2.2.

###### Proof.

If then there exists a Carathéodory function of the form (2.1) such that , where . Following the same method as used in Theorem 2.1 and noting that , a simple computation in (2.4) shows that

(2.16) |

where and . Since and , then . Taking modulus on the both sides of (2.16) and then applying triangle inequality and writing , it follows that

where we have also used the fact . Let where and . For simplicity, by writing we obtain

(2.17) |

where and

Thus we need to find the maximum value of over the rectangular cube .

We first find the maximum value of on the boundary of , i.e on the six faces of the rectangular cube . As before, let and . By elementary calculus it is not very difficult to prove that

On the face , we have where . As in the proof of Theorem 2.1, one can verify that in the interior of (otherwise, one can simply proceed to find maximum value at an interior point of , where , as in ). Suppose that has the maximum value at an interior point of . Then at such point and . From (for points in the interior of ), it follows that

(2.18) |

By substituting the above value of given in (2.18) in the relation and further computation (a long and laborious calculation) gives

This equation has exactly two real roots in