General Univalence Criterion Associated with the n th Derivative

and Applied Analysis 3 where K n! n − 1 ( m M2 − n−1 ∑ k 2 k − 1 k! |αk| ) , αk d dzk ( z g z − z f z ∣∣∣∣ z 0 , 2.3 andM sup{|g z | : z ∈ U}, then f z is univalent in U. Proof. If we put h z d dzn ( z f z − z g z ) , 2.4 then the function h is analytic in U and, by integration from 0 to z, we get dn−1 dzn−1 ( z f z − z g z ) αn−1 ∫z 0 h u1 du1. 2.5 Integrating both sides of the previous equation n − 1 -times from 0 to z gives z f z − z g z n−1 ∑ k 1 αk k! z ∫z 0 dun ∫un 0 dun−1 · · · ∫u3 0 du2 ∫u2 0 h u1 du1. 2.6


Introduction
Let A denote the class of functions of the following form: a n z n , 1.1 which are normalized analytic in the open unit disk U : {z : |z| < 1}.In 1 , Aksentev proved that the condition or equivalently Re f 2 z /z 2 f z ≥ 1/2, for z ∈ U, is sufficient for f z ∈ A to be univalent in U.By virtue of the aforementioned result of Aksentev, the class of functions defined by 1.2 was extensively studied by Obradović  In this work, we introduce a univalence criteria defined by the conditions f z / 0 for 0 < |z| < 1 and where f z is normalized analytic in U and β k d k /dz k z/f z | z 0 , n ∈ {3, 4, . ..}.The sharpness occurs for the Koebe function.Indeed, all functions satisfying the condition 1.5 are univalent in U and the bound 1 in the inequality is best possible for univalence.Letting n 2 in 1.5 gives the univalence criterion defined by 1.4 .Some special cases and examples for functions satisfying 1.5 are given.

Sufficient Conditions for Univalence
Let us prove the following theorem.Theorem 2.1.Let f z ∈ A with f z / 0 for 0 < |z| < 1 and let g z ∈ A be bounded in U and satisfy For any n ∈ {3, 4, . ..}, if where Proof.If we put then the function h is analytic in U and, by integration from 0 to z, we get Integrating both sides of the previous equation n − 1 -times from 0 to z gives Thus, we have where Next, for n 3, we have and for n 4,

2.10
In general, for n ∈ {3, 4, . ..}, 12 and so for z 1 , z 2 ∈ U and z 1 / z 2 .If z 1 / z 2 , then g z 1 / g z 2 , and it follows, from 2.7 and 2.13 , that where The result is sharp, where equality occurs for the Koebe function k z z/ 1 − z 2 and also for functions of the following form:

2.16
Proof.Setting g z z in Theorem 2.1 immediately yields 2.15 .To show that the result is sharp for n ≥ 3, we consider

2.18
Letting 0 in 2.17 and 2.18 implies, respectively, that d n /dz n z/f z 0 and

2.19
This satisfies the equality in 2.15 , because for x ∈ R and n ≥ 3, an application of the binomial theorem gives and so

2.21
Choosing x 1/ n − 2 in assertion 2.21 gives the equality.However, for every > 0, we have Hence f is not univalent in U and the result is sharp.Moreover it can be easily checked that the equality in 2.15 holds for the given functions and the proof is complete.

Special Cases and Examples
Letting n 2 in inequality 2.15 gives the univalence criterion defined by 1.4 , which is due to Yang and Liu 7 .Next, we reduce the result for some values of n by computing the corresponding values of β k in terms of the coefficients.More precisely, for n 3 and n 4, Corollary 2.2 reduces at once to the following two remarks.
Then f z is univalent in U.The bound in 3.1 is best possible, where equality occurs for the Koebe function and for functions of the following form: Abstract and Applied Analysis 7 Proof.The result follows from taking n 3 in Corollary 2.2 and that Then f z is univalent in U.The bound in 3.3 is best possible, where equality occurs for the Koebe function and also for functions of the following form: To understand the behavior of the extremal functions for our criterion 2.15 , let us consider, for example, f z z/ 1 − 1/2 z 3 , which is an extremal function for the case n 4. Figures 1 a and 1 b show the images of the unit circle under the functions f z and g z z/ 1 − 1/2 z 3.05 , respectively.If we restrict the images around the cusps as shown in Figures 1 c and 1 d , we see that the image of g is a curve that intersects itself in some purely real point u.This means that there are two different points z 1 and z 2 that lie on the unit circle such that g z 1 g z 2 u.In fact, each purely real point lies inside the closed curve of Figures 1 c and 1 d which is an image for two different points in U having the same modulus but different arguments.However, we cannot find such points for the function f, and this interprets why f is an extremal function for univalence, since the closed curve of Figure 1 d vanishes whenever the power in the function g approaches to 3 as shown in Figure 1 c .From Corollary 2.2, we have the following.
Proof.In view of 3.5 and by simple computation we have
Abstract and Applied AnalysisObradović et al. 5 , and others.Afterwards, Nunokawa et al. 6 proved for f z ∈ A with f z / 0 when 0 < |z| < 1 that 1| ≤ 1 for z ∈ U, and hence f z is univalent in U. Later, Yang and Liu 7 extended this result for f z ∈ A: and Ponnusamy 2, 3 , Ozaki and Nunokawa 4 , 2 with f z / 0 when 0 < |z| < 1 implies that f z is univalent in U and the bound 2 is best possible for univalence.This result was also given first in the preprint of reports of the Department of Mathematics, University of Helsinki: M. Obradović, S. Ponnusamy, New criteria, and distortion theorems for univalent functions, Preprint 190, June 1998.Later, under the same name, the paper was published in Complex Variables Theory Application see 3 .Corresponding to the functions defined by 1.4 , Yang and Liu in 7 studied a class of analytic univalent functions f z satisfying | z/f z | ≤ β 0 < β ≤ 2 and denoted by S β .The class S β is extensively studied in the recent years see 2, 3, 8-10 .