On an extension of Nunokawa’s lemma

Jack’s Lemma says that if f(z) is regular in the disc |z|≤r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|\le r$$\end{document}, f(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(0)=0$$\end{document}, and |f(z)| assumes its maximum at z0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0$$\end{document} on the circle |z|=r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|=r$$\end{document}, then z0f′(z)0/f(z0)≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0f'(z)_0/f(z_0)\ge 1$$\end{document}. This Lemma was generalized in several directions. In this paper we consider an improvement of some first author’s results of this type.


Introduction
Let H be the family of analytic functions in the region D = {z : |z| < 1} on the complex plane C. Let A ⊂ H denote the set of all functions f (z) that are analytic in D with the series representation f (z) = z + ∞ n=2 a n z n , z ∈ D. (1.1) Also by S we means a subfamily of the set A which contains univalent functions. We next denote by P the class of analytic functions p(z) which are normalized by  such that Re{ p(z)} > 0, z ∈ D. Furthermore, by using the set P, let S * and C denote the families of starlike and convex functions in D which are defined as The above subfamilies of the set S are among the most studied families. Recall also, for some g ∈ S * and some α ∈ (−π/2, π/2), then f is said to be close-to-convex in D and denoted by f ∈ K. An univalent function f ∈ A belongs to K if and only if the complement E of the image-region F = { f (z) : |z| < 1} is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays). We note that if g (z) = z, then the class K reduces to the set R of bounded turning functions. Jack's Lemma [2], says that if f (z) is regular in the disc |z| ≤ r , f (0) = 0, and | f (z)| assumes its maximum at z 0 on the circle |z| = r , then z 0 f (z) 0 / f (z 0 ) ≥ 1. Now we will consider a lemma, which is a small extension of Jack's Lemma.
Then ϕ(z) is analytic in D and |ϕ(z)| takes its local maximum value at the point z = z 0 on the circular arc z = |z 0 |e iθ , arg z 0 −ε < θ < arg z 0 +ε. When z moves on the circle |z| = |z 0 | with positive direction, then arg ϕ(z) is increasing at z = z 0 . Therefore, we have d arg ϕ(|z 0 |e iθ ) d` From the hypothesis, we have

This shows
Assume there exist two points z 1 , z 2 ∈ D such that |z 1 | = |z 2 | and for some ε > 0, then we have where p 1/α (z 1 ) = ia, a > 0, and Proof Let us check the first case arg p(z 1 ) = πα/2 and put then it follows that From the hypothesis, we have Therefore, we have and This shows that |ϕ(z)| takes its local maximum value at the point z = z 1 in the the domain N ε (z 1 ) and applying Lemma 1.1, we have where k ≥ 1 and q(z 1 ) = ia and a > 0. This shows that z 1 q (z 1 ) is a negative real number. This gives On the other hand, we have and putting z = |z 1 |e iθ in the section arg z 1 − ε < θ < arg z 1 + ε, then arg{q(z 1 )} takes its local maximum π/2 at the point θ = arg{z 1 } in this section and so, we have Re Therefore, we have where p 1/α (z 1 ) = ia, a > 0. This completes the proof of (1.13).
For the next case arg p(z 2 ) = −πβ/2 let us put then it follows that and so From the hypothesis, we have for some ε > 0 and arg ρ(z 2 ) = − π 2 .
This shows that |ψ(z)| takes its local maximum value at the point z = z 2 in the the domain N ε (z 2 ) and applying Lemma 1.1, we have where k ≥ 1 and ρ(z 2 ) = −ib with b > 0. This shows that z 2 ρ (z 2 ) is a negative real number. This gives On the other hand, we have and putting z = |z 2 |e iθ in the section arg z 2 − ε < θ < arg z 2 + ε, then arg{ρ(z 2 )} takes its local minimum −π/2 at the point θ = arg{z 2 } in this section and so, we have Re This completes the proof of (1.14). [4,5].

Corollary 1.3 Under the assumptions of Theorem 1.2 we have
Corollary 1.3 implies that the length of image curve under zp (z)/ p(z) of the circular arc |z| = |z 1 | = |z 2 | from z 2 to z 1 is great or equal to (α + β)k.
Proof From Lemma 1.1 we have On the other hand and applying (1.15), we have where arg z 1 = θ 1 , arg z 2 = θ 2 and θ 2 < θ 1 . This shows that 0 < α + β < 1 which implies 0 < α < 1 and 0 < β < 1 or In [8, p.54] we can find the following result for the convolution of power series ∞ n=0 a n z n * ∞ n=0 b n z n = ∞ n=0 a n b n z n .

Lemma 1.5 If g(z) ∈ C, f (z) ∈ S * and F(z) is analytic in D, then for all z
where co A denotes the closed convex hull of A.
The differential subordination theory provides another method od proving this type of results. For two functions f (z) and g(z) analytic in D, we say that f f (z) is subor-dinate to g(z), written by f (z) ≺ g (z), if there exists a function w(z), analytic in D, with w(0) = 0 and |w(z)| < 1 such that f (z) = g(w(z)).
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