Abstract
For f analytic in the unit disk \(\mathbb {D}\), of the form \(f(z)=z^p+\cdots \), we consider some consequences of strongly starlikeness of \(f^{(p-1)}(z)/p!\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
We denote by \(\mathcal {H}\) the class of functions f(z) which are holomorphic in the open unit disc \({\mathbb {D}}=\{z\in \mathbb {C}: |z|<1\}\). Denote by \({\mathcal {A}}_p\), \(p\in \mathbb {N}=\{1,2,\ldots \} \), the class of functions \(f(z)\in \mathcal {H}\) given by
Lemma 1.1
[2, Theorem 5] If \(f(z)\in {\mathcal {A}}_p\), then for all \(z\in {\mathbb {D}}\), we have
In this paper we consider a generalization of the above result. In Lemma 1.1 we have assumed that \(zf^{(p)}(z)/f^{(p-1)}(z)\) 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 \(\alpha ,\beta \) does an analytic function of the form (1.1) satisfy
Recall that if \(f(z)\in {\mathcal {A}}_p\) and
then \(f^{(p-1)}(z)/p!\in {\mathcal {A}}_1\) is univalent in \({\mathbb {D}}\) and \(f^{(p-1)}(z)/p!\) is called a starlike function. If \(f(z)\in {\mathcal {A}}_p\), \(\gamma \in (0,1]\), and
then \(f^{(p-1)}(z)/p!\) is called a strongly starlike function of order \(\gamma \) and such functions we consider in the paper. This class for the case \(p=1\) was introduced by Brannan and Kirwan [1]. Also, if \(f(z)\in {\mathcal {A}}_p\) satisfies (1.3), then f(z) is called p-valently strongly starlike function of order \(\gamma \). For the proof of main result we need the following lemma.
Lemma 1.2
[3] Let \(q(z)=1+\sum _{n\ge m}^{\infty }c_nz^n\), \(c_m\ne 0\) be analytic function in \(|z|<1\) with \(q(0)=1\), \(q(z)\ne 0\). If there exists a point \(z_0\), \(|z_0|<1\), such that
and
for some \(\beta >0\), then we have
for some \(k\ge m(a+a^{-1})/2\ge m\), where
2 Main results
For given \(0<\beta _{s-1}\le 1\) let us consider the number
where
Notice that if \(0<\beta _{s-1}\le 1\), then \(0<\beta _s\le 1\) too because from (2.1), (2.2), we have
Therefore, if we have a number \(\beta _1\in (0,1]\), then from (2.1), we can find a sequence \(\beta _p,\beta _{p-1},\ldots ,\beta _2,\beta _1\), such that
Theorem 2.1
Let \(f(z)\in {\mathcal {A}}_p\), \(p\ge 2\). For given \(\beta _{p-1}\in (0,1]\) there exists \(\beta _{p}\in (0,1]\) of the form (2.1) such that for all \(z\in {\mathbb {D}}\), we have
Proof
Let us put
Then it follows that
and
and so
If there exists a point \(z_0\in {\mathbb {D}}\) such that
then from Lemma 1.2, we have
for some real k with \(k\ge (a+a^{-1})/2\ge 1\). For the case \(\arg \{q_1(z_0)\}=\pi \beta _{p-1}/2\), we have
where \(\left\{ q_1(z_0)\right\} ^{1/\beta _{p-1}}=ia\) and a is a positive real number. Applying Lemma 1.2 we obtain
From (2.1), we can see that
This contradicts hypothesis in (2.4).
For the case \(\arg \{q_1(z_0)\}=-\pi \beta _{p-1}/2\), applying the same method as the above, gives
This also contradicts hypothesis in (2.4) and therefore, we have
This completes the proof. \(\square \)
Let us go to next step and define the function
and applying the same method as the above, we have the following theorem.
Theorem 2.2
Let \(f(z)\in {\mathcal {A}}_p\), \(p\ge 2\), \(0<\beta _2\le 1\) and suppose that
Then we have
where \(\beta _{p-2}\) we obtain from \(\beta _{p-1}\) using formula (2.1). Furthermore,
and where \(\beta _{p-1}\) we obtain from \(\beta _{p}\) using formula (2.1) too.
Applying the same step as the above and under the hypothesis of Theorem 2.1, we have the following theorem
Theorem 2.3
Let \(f(z)\in {\mathcal {A}}_p\), \(p\ge 2\). For given \(\beta _{1}\in (0,1]\) there exist \(\beta _{k}\in (0,1]\), \(k=2,\ldots ,p\), of the form (2.1) such that for all \(z\in {\mathbb {D}}\), we have
Furthermore
It is easy to see that Theorem 2.3 holds for the case \(\beta _{p}=\beta _{p-1}=\cdots =\beta _{1}=1\) and then Theorem 2.3 becomes Lemma 1.1 and in this sense Theorem 2.3 improves Lemma 1.1.
Corollary 2.4
Let \(f(z)\in {\mathcal {A}}_p\), \(p\ge 2\). If \(\beta _1\in (0,1]\) and \(\beta _{k}\in (0,1]\), \(k=2,\ldots ,p\) are of the form (2.1), then for all \(k=1,\ldots ,p-1\) and for all \(z\in {\mathbb {D}}\), we have
Proof
For given \(\beta _1\in (0,1]\) there exist \(\beta _{k}\in (0,1]\), \(k=2,\ldots ,p\), of the form (2.1) such that
where
Therefore, from (2.11), we have
\(\square \)
Corollary 2.5
If \(f(z)\in {\mathcal {A}}_p\), \(p\ge 2\), then for all \(\gamma \in (0,1]\) and for all \(k\in \{1,\ldots ,p\}\) and for all \(s\in \{k,\ldots ,p-1\}\), and for all \(z\in {\mathbb {D}}\), we have
From the properties of the sequence (2.1) we have following corollary.
Corollary 2.6
Let \(f(z)\in {\mathcal {A}}_p\), \(p\ge 2\), \(0<\beta _{p}\le 1\) and suppose that
Then we have
Theorem 2.7
Let \(\beta =\alpha +(2/\pi )\tan ^{-1}\alpha \) and \(f(z)\in {\mathcal {A}}_p\), \(p\ge 2\). Suppose also that
Then we have
where \(\beta _{1}\) is described in (2.1) with \(\beta _{p}=\alpha +\beta \).
Proof
If
in \(|z<|z_0|\) and
then for the first case in (2.15), from Lemma 1.2, we have
for some \(k\ge 1\), This gives
This contradicts hypothesis (2.12). In the second case in (2.15), applying the same method as in the first case, we obtain
This also contradicts hypothesis (2.12). So (2.14) holds in the whole unit disc \({\mathbb {D}}\). From (2.12) and (2.14), we have
References
Brannan, D.A., Kirwan, W.E.: On some class of bounded univalent functions. J. Lond. Math. Soc. 2(1), 1431–1443 (1969)
Nunokawa, M.: On the theory of multivalent functions. Tsukuba J. Math. 11(2), 273–286 (1987)
Nunokawa, M.: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad. Ser. A 69(7), 234–237 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Nunokawa, M., Sokół, J. On a successive property of strongly starlikeness for multivalent functions. Rend. Circ. Mat. Palermo, II. Ser 69, 939–945 (2020). https://doi.org/10.1007/s12215-019-00442-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-019-00442-z