# A CRITERION FOR BOUNDED FUNCTIONS

• Nunokawa, Mamoru (University of Gunma) ;
• Owa, Shigeyoshi (Kinki University) ;
• Sokol, Janusz (Department of Mathematics Rzeszow University of Technology)
• Published : 2016.01.31
• 93 12

#### Abstract

We consider a sufficient condition for w(z), analytic in ${\mid}z{\mid}$ < 1, to be bounded in ${\mid}z{\mid}$ < 1, where $w(0)=w^{\prime}(0)=0$. We apply it to the meromorphic starlike functions. Also, a certain Briot-Bouquet differential subordination is considered. Moreover, we prove that if $p(z)+zp^{\prime}(z){\phi}(p(z)){\prec}h(z)$, 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

#### References

1. H. Al-Amiri and P. T. Mocanu, Some simple criteria of starlikeness and convexity for meromorphic functions, Mathematica 37(60) (1995), no. 1-2, 11-21.
2. S. Fukui and K. Sakaguchi, An extension of a theorem of S. Ruscheweyh, Bull. Fac. Ed. Wakayama Univ. Natur. Sci. 29 (1980), 1-3.
3. I. S. Jack, Functions starlike and convex of order ${\alpha}$, J. London Math. Soc. 3 (1971), 469-474.
4. D. I. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc. 52 (1975), 191-195. https://doi.org/10.1090/S0002-9939-1975-0374403-3
5. S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Series of Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York/Basel 2000.
6. M. Nunokawa, On properties of non-Caratheodory functions, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 6, 152-153. https://doi.org/10.3792/pjaa.68.152
7. M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 7, 234-237. https://doi.org/10.3792/pjaa.69.234
8. M. Nunokawa and J. Soko l, Strongly gamma-starlike functions of order alpha, Ann. Univ. Mariae Curie-Sk lodowska, LXVII/2 (2013), 43-51.
9. M. Nunokawa and J. Soko l, New conditions for starlikeness and strongly starlikeness of order alpha, Houston Journal of Mathematics, in press.
10. T. N. Shanmugam, S. Sivasubramanian, and H. M. Srivastava, On sandwich theorems for some classes of analytic functions, Int. J. Math. Math. Sci. 2006 (2006), Article ID 29684, 1-13.

#### Cited by

1. NOTES ON THE PAPER "A CRITERION FOR BOUNDED FUNCTIONS" [BULL. KOREAN MATH. SOC. 53 (2016), NO. 1, 215-225] vol.53, pp.6, 2016, https://doi.org/10.4134/BKMS.b160197