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

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

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.

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
     author = {Nunokawa, Mamoru and Sok\'o{\l}, Janusz and Thomas, Derek K.},
     title = {On {Ozaki's} condition for \protect\emph{p}-valency},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {382--386},
     publisher = {Elsevier},
     volume = {356},
     number = {4},
     year = {2018},
     doi = {10.1016/j.crma.2018.02.007},
     language = {en},
     url = {}
AU  - Nunokawa, Mamoru
AU  - Sokół, Janusz
AU  - Thomas, Derek K.
TI  - On Ozaki's condition for p-valency
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 382
EP  - 386
VL  - 356
IS  - 4
PB  - Elsevier
UR  -
DO  - 10.1016/j.crma.2018.02.007
LA  - en
ID  - CRMATH_2018__356_4_382_0
ER  - 
%0 Journal Article
%A Nunokawa, Mamoru
%A Sokół, Janusz
%A Thomas, Derek K.
%T On Ozaki's condition for p-valency
%J Comptes Rendus. Mathématique
%D 2018
%P 382-386
%V 356
%N 4
%I Elsevier
%R 10.1016/j.crma.2018.02.007
%G en
%F CRMATH_2018__356_4_382_0
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.

[1] Noshiro, K. On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. Jap., Volume 2 (1934–1935) no. 1, pp. 129-135

[2] Nunokawa, M. A note on multivalent functions, Tsukuba J. Math., Volume 13 (1989) no. 2, pp. 453-455

[3] Nunokawa, M. On properties of non-Carathéodory functions, Proc. Jpn. Acad., Ser. A, Volume 68 (1992) no. 6, pp. 152-153

[4] Ozaki, S. On the theory of multivalent functions, Sci. Rep. Tokyo Bunrika Daigaku, Sect. A., Volume 2 (1935), pp. 167-188

[5] Pommerenke, Ch. On close to-convex functions, Trans. Amer. Math. Soc., Volume 114 (1965) no. 1, pp. 176-186

[6] Sakaguchi, K. On certain univalent mapping, J. Math. Soc. Jpn., Volume 11 (1959), pp. 72-75

[7] Umezawa, T. Multivalently close-to-convex functions, Proc. Amer. Math. Soc., Volume 8 (1957) no. 5, pp. 869-874

[8] Warschawski, S. On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc., Volume 38 (1935), pp. 310-340

Cited by Sources: