Integral operators that preserve the subordination

Abstract

Let $H(U)$ be the space of all analytic functions in the unit disk $U$ and let $K \subset H(U)$. For the operator $A_{\beta,\gamma} : K \longrightarrow H(U)$ defined by $$ A_{\beta,\gamma}(f)(z) = [\frac{z^\gamma}{\beta + \gamma} \int_{0}^{z} f^\beta (t)t^{\gamma-1} dt]^{1/\beta} $$ and $\beta,\gamma \in C$, we determined conditions on g(z), $\beta and \gamma$ such that $$ z[\frac{z}{f(z)]^\beta \prec z[\frac{z}{g(z)]^\beta implies z[\frac{z}{A_{\beta,\gamma}(f)(z)]^\beta \prec z[\frac{z}{A_{\beta,\gamma}(g)(z)]^\beta $$ and we presented some particular cases of our main result.

Keywords

• Differential subordination;
• integral operator;
• subordination chain;
• univalent function

File

Download PDF

References

1. Indian J. Pure Appl. Math. v.28 On a special integral subordination T. Bulboaca
2. Dokl. Akas. Nauk. SSSR v.42 On the majorization principle in function theory G.M. Goluzin
3. Proc. Amer. Math. Soc. v.52 Subordination by convex functions D.J. Hallenbeck;S. Ruscheweyh
4. Michigam Math. J. v.28 Differential subordinations and univalent functions S.S. Miller;P.T. Mocanu
5. J. Differential Equations v.67 Univalent solutions of Briot-Bouquet differentialequations S.S. Miller;P.T. Mocanu
6. Fractional Calculus and their Applications Integral operatirs on certain classes of analyti:functions, Univalent Functions S.S. Miller;P.T. Mocanu
7. Trans. Amer. Math. Soc. v.283 Subordination-preserving integral operators S.S. Miller;P.T. Mocanu;M.O. Reade
8. Vanderhoeck and Ruprecht "Univalent Functions" Ch. Pommerenke
9. Duke Math. J. v.37 Some remarks on complex maps of the unit disk T.J. Suffridge