Comptes Rendus
no. 4
Mathematical analysis
On Ozaki's condition for p-valency

Presented by: the Editorial Board

Mamoru Nunokawa1; Janusz Sokół2; Derek K. Thomas3
1 University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan
2 University of Rzeszów, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland
3 Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK
Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 382-386.

Let f be an analytic function in a convex domain D⊂C. A well-known theorem of Ozaki states that if f is analytic in D, and is given by f(z)=zp+∑n=p+1∞anzn for z∈D, and

Re{eiαf(p)(z)}>0,(z∈D),
for some real α, then f is at most p-valent in D. Ozaki's condition is a generalization of the well-known Noshiro–Warschawski univalence condition. The purpose of this paper is to provide some related sufficient conditions for functions analytic in the unit disk D={z∈C:|z|<1} to be p-valent in D, and to give an improvement to Ozaki's sufficient condition for p-valence when z∈D.

Soit f une fonction analytique dans un domaine D⊂C. Un théorème bien connu d'Ozaki affirme que, si f est analytique dans D, donnée par f(z)=zp+∑n=p+1∞anzn pour z∈D et

Re{eiαf(p)(z)}>0,(z∈D),
pour un réel α, alors f est au plus p-valuée dans D. La condition d'Ozaki est une généralisation d'une condition de Noshiro–Warschawski pour qu'une fonction soit univaluée, également bien connue. Notre propos ici est de fournir des conditions suffisantes pour que des fonctions analytiques dans le disque unité D={z∈C;|z|<1} soient p-valuées dans D et d'améliorer la condition suffisante d'Ozaki correspondante quand z∈D.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.02.007

Mamoru Nunokawa 1; Janusz Sokół 2; Derek K. Thomas 3

1 University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan
2 University of Rzeszów, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland
3 Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK 
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Mamoru Nunokawa; Janusz Sokół; Derek K. Thomas. On Ozaki's condition for p-valency. Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 382-386. doi : 10.1016/j.crma.2018.02.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.02.007/

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