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 A criterion for bounded functions Bull. Korean Math. Soc. 2016 Vol. 53, No. 1, 215-225 https://doi.org/10.4134/BKMS.2016.53.1.215Printed January 31, 2016 Mamoru Nunokawa, Shigeyoshi Owa, and Janusz Sok\'{o}\l University of Gunma, Kinki University, Rzesz\'{o}w University of Technology Abstract : We consider a sufficient condition for $w(z)$, analytic in $|z|<1$, to be bounded in $|z|<1$, where $w(0)=w'(0)=0$. We apply it to the meromorphic starlike functions. Also, a certain Briot-Bouquet differential subordination is considered. Moreover, we prove that if $p(z)+zp'(z)\phi(p(z))\prec h(z)$, then $p(z)\prec h(z)$, where $h(z)=[(1+z)(1-z)]^{\alpha}$, under some additional assumptions on $\phi(z)$. Keywords : analytic, meromorphic, convex, starlike, univalent, Nunokawa's lemma, Briot-Bouquet, differential subordination MSC numbers : Primary 30C45; Secondary 30C80 Downloads: Full-text PDF