On Generalized Integral Operator Based on Salagean Operator

• Journal title : Kyungpook mathematical journal
• Volume 48, Issue 3,  2008, pp.359-366
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2008.48.3.359
Title & Authors
On Generalized Integral Operator Based on Salagean Operator
Al-Kharsani, Huda Abdullah;

Abstract
Let A(p) be the class of functions $\small{f\;:\;z^p\;+\;\sum\limits_{j=1}^{\infty}a_jz^{p+j}}$ analytic in the open unit disc E. Let, for any integer n > -p, $\small{f_{n+p-1}(z)\;=\;z^p+\sum\limits_{j=1}^{\infty}(p+j)^{n+p-1}z^{p+j}}$. We define $\small{f_{n+p-1}^{(-1)}(z)}$ by using convolution * as $\small{f_{n+p-1}\;*\;f_{n+p-1}^{-1}=\frac{z^p}{(1-z)^{n+p}}$. A function p, analytic in E with p(0) = 1, is in the class $\small{P_k(\rho)}$ if $\small{{\int}_0^{2\pi}\|\frac{Re\;p(z)-\rho}{p-\rho}\|\;d\theta\;\leq\;k{\pi}}$, where $\small{z=re^{i\theta}}$, $\small{k\;\geq\;2}$ and $\small{0\;{\leq}\;\rho\;{\leq}\;p}$. We use the class $\small{P_k(\rho)}$ to introduce a new class of multivalent analytic functions and define an integral operator $\small{L_{n+p-1}(f)\;\;=\;f_{n+p-1}^{-1}\;*\;f}$ for f(z) belonging to this class. We derive some interesting properties of this generalized integral operator which include inclusion results and radius problems.
Keywords
convolution;integral operator;functions with positive real part;convex functions;
Language
English
Cited by
References
1.
M. Acu, A preserving property of the generalized Bernardi integral operator, General Mathematics, 12(3)(2004), 67-71.

2.
M. K. Aouf and B. A. Al-amri, On certain fractional operators for certain subclasses of prestarlike functions defined by Salagean operator, Journal of Fractional Calculus, 22(2002), 47-56.

3.
K. Inayat Noor, On subclasses of close-to-convex functions of higher order, Internal. J. Math. Sc., 15(1992), 279-290.

4.
K. S. Padmanabhan and R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math., 31(1975), 311-323.

5.
S. Ponnusamy, Differential subordination and Bazievic functions, preprint.

6.
G. Salagean, Subclasses of univalent functions, Complex Analysis, Fifth Roumanian-Finish Seminar, Lecture Notes in Mathematics, 1013, Springer-Verlag, 1983, 362-372.

7.
R. Singh and S. Sing, Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., 106(1989), 145-152.