Vol.08 No.03(2017), Article ID:74883,5 pages
New Result for Strongly Starlike Functions
R. O. Ayinla1, T. O. Opoola2
1Department of Statistics and Mathematical Sciences, Kwara State University, Malete, Nigeria
2Department of Mathematics, University of Ilorin, Ilorin, Nigeria
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: January 12, 2017; Accepted: March 21, 2017; Published: March 24, 2017
In this paper, using Salagean differential operator, we define and investigate a new subclass of univalent functions. We also establish a characterization property for functions belonging to the class.
Strongly Starlike Functions, Strongly Convex Functions, Salagean Differential Operator
Let be the class of functions of the form
which are analytic in the unit disk. A function is said to be starlike of order if and only if
We denote by the subclass of consisting of functions which are starlike of order in.
Also, a function is said to be convex of order if and only if
We denote by the subclass of consisting of functions which are convex of order in.
then is said to be strongly starlike of order and type in, denoted by  .
then is said to be strongly convex of order and type in, denoted by  .
The following lemma is needed to derive our result for class.
Lemma (1)     . Let a function be analytic in , if there exists a point such that
and with, then
Definition 1. A function is said to be in the class if
When then is the class studied by  .
Definition 2. For functions the Salagean differential operator  is
The main focus of this work is to provide a characterization property for the class of functions belonging to the class.
2. Main Result
Theorem 1. If satisfies
for some then
Taking the logarithmic differentiation in both sides of Equation (8), we have
Multiply Equation (9) through by, to get
Multiply Equation (10) by to obtain
Multiply Equation (11) through by 2 and divide through by to give
Multiplying Equation (12) by, and further simplifica-
tion, we obtain
If a point which satisfies and
then by lemma 
It implies that
is a minimum
Therefore, we have that
which contradicts the condition of the theorem.
Hence, it is concluded from lemma  that
The authors wish to thank the referees for their useful suggestions that lead to improvement of the quality of the work in this paper.
Cite this paper
Ayinla, R.O. and Opoola, T.O. (2017) New Result for Strong- ly Starlike Functions. Applied Mathematics, 8, 324-328. https://doi.org/10.4236/am.2017.83027
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