Mathematical analysis
On Ozaki's condition for p-valency
Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 382-386.

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

Re{eiαf(p)(z)}>0,(zD),
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={zC:|z|<1} to be p-valent in D, and to give an improvement to Ozaki's sufficient condition for p-valence when zD.

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

Re{eiαf(p)(z)}>0,(zD),
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={zC;|z|<1} soient p-valuées dans D et d'améliorer la condition suffisante d'Ozaki correspondante quand zD.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.02.007
Nunokawa, Mamoru 1; Sokół, Janusz 2; Thomas, Derek K. 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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Nunokawa, Mamoru; Sokół, Janusz; Thomas, Derek K. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.02.007/

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