# Radius of Convexity for Some Integral Operators on Function Spaces

## Abstract

In this paper, we study the radius of convexity of the following integral operators

$$I_{n}^{{\gamma_{i} }} \left( {f_{1} , \ldots ,f_{n} } \right) = F\left( z \right) \, : = \mathop \int \limits_{0}^{z} \mathop \prod \limits_{i = 1}^{n} \left( {f_{i}^{'} \left( t \right)} \right)^{{\gamma_{i} }} {\text{d}}t$$

and

$$J_{n}^{{\gamma_{i} ,\lambda_{j} }} \left( {f_{1} , \ldots ,f_{n} ;g_{1} , \ldots ,g_{m} } \right) = J\left( z \right)\text{ := }\mathop \int \limits_{0}^{z} \mathop \prod \limits_{i = 1}^{n} \left( {f_{i}^{'} \left( t \right)} \right)^{{\gamma_{i} }} \mathop \prod \limits_{j = 1}^{m} \left( {\frac{{g_{j} \left( z \right)}}{z}} \right)^{{\lambda_{j} }} {\text{d}}t,$$

where $$f_{i} \left( {1 \le i \le n} \right)$$ and $$g_{j } \left( {1 \le j \le m} \right)$$ belong to some certain subclasses of analytic functions. Also, we investigate the univalency of

$$J_{n}^{{\gamma_{i} ,\lambda_{i} }} \left( {f_{1} , \ldots ,f_{n} ;\;g_{1} , \ldots ,g_{n} } \right)$$

under some conditions.

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## Acknowledgements

This paper forms a part of Ph.D. thesis of the first author.

Authors