On a successive property of strongly starlikeness for multivalent functions
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On a successive property of strongly starlikeness for multivalent functions Mamoru Nunokawa1 · Janusz Sokół2 Received: 31 May 2019 / Accepted: 6 August 2019 / Published online: 31 August 2019 © The Author(s) 2019
Abstract For f analytic in the unit disk D, of the form f (z) = z p +· · · , we consider some consequences of strongly starlikeness of f ( p−1) (z)/ p!. Keywords Starlike · Strongly starlike · Multivalent Mathematics Subject Classification Primary 30C45; Secondary 30C80
1 Introduction We denote by H the class of functions f (z) which are holomorphic in the open unit disc D = {z ∈ C : |z| < 1}. Denote by A p , p ∈ N = {1, 2, . . .}, the class of functions f (z) ∈ H given by ∞ f (z) = z p + an z n , (z ∈ D). (1.1) n= p+1
Lemma 1.1 [2, Theorem 5] If f (z) ∈ A p , then for all z ∈ D, we have z f (k) (z) z f ( p) (z) > 0 ⇒ ∀k ∈ {1, . . . , p − 1} : Re > 0. Re f ( p−1) (z) f (k−1) (z)
(1.2)
In this paper we consider a generalization of the above result. In Lemma 1.1 we have assumed that z f ( p) (z)/ f ( p−1) (z) lies in the right half-plane while in this paper we work with a sector. The problem we solve here is: for what values of α, β does an analytic function of the form (1.1) satisfy
B
Janusz Sokół [email protected] Mamoru Nunokawa [email protected]
1
University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba 260-0808, Japan
2
Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, 35-310 Rzeszow, Poland
123
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M. Nunokawa, J. Sokół
z f ( p) (z) πα < arg 2 f ( p−1) (z)
⇒
∀k ∈ {1, . . . , p − 1} :
z f (k) (z) πβ ? < arg 2 f (k−1) (z)
Recall that if f (z) ∈ A p and Re
z f ( p) (z) f ( p−1) (z)
> 0 (z ∈ D),
then f ( p−1) (z)/ p! ∈ A1 is univalent in D and f ( p−1) (z)/ p! is called a starlike function. If f (z) ∈ A p , γ ∈ (0, 1], and z f ( p) (z) πγ arg ( p−1) < , z ∈ D, f 2 (z)
(1.3)
then f ( p−1) (z)/ p! is called a strongly starlike function of order γ and such functions we consider in the paper. This class for the case p = 1 was introduced by Brannan and Kirwan [1]. Also, if f (z) ∈ A p satisfies (1.3), then f (z) is called p-valently strongly starlike function of order γ . For the proof of main result we need the following lemma. n Lemma 1.2 [3] Let q(z) = 1 + ∞ n≥m cn z , cm = 0 be analytic function in |z| < 1 with q(0) = 1, q(z) = 0. If there exists a point z 0 , |z 0 | < 1, such that |arg {q(z)} |
0, then we have 2ik arg {q(z 0 )} z 0 q (z 0 ) = , q(z 0 ) π for some k ≥ m(a + a −1 )/2 ≥ m, where {q(z 0 )}1/β = ±ia, and a > 0.
2 Main results For given 0 < βs−1 ≤ 1 let us consider the number βs = βs−1 +
βs−1 n(βs−1 ) sin[π(1 − βs−1 )/2] 2 tan−1 s = 2, 3, . . . , p, π sm(βs−1 ) + βs−1 n(βs−1 ) cos[π(1 − βs−1 )/2], (2.1)
where m(βs−1 ) = (1 + βs−1 )(1+βs−1 )/2 , and n(βs−1 ) = (1 − βs−1 )(1−βs−1 )/2 .
123
(2.2)
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