AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2017.83027AM-74883ArticlesPhysics&Mathematics New Result for Strongly Starlike Functions R.O. Ayinla1*T.O. Opoola2Department of Mathematics, University of Ilorin, Ilorin, NigeriaDepartment of Statistics and Mathematical Sciences, Kwara State University, Malete, Nigeria* E-mail:rasheed.ayinla@kwasu.edu.ng(ROA);030320170803324328January 12, 2017Accepted: March 21, March 24, 2017© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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
1. Introduction

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.

If satisfies

then is said to be strongly starlike of order and type in, denoted by [1] .

If satisfies

then is said to be strongly convex of order and type in, denoted by [1] .

The following lemma is needed to derive our result for class.

Lemma (1) [2] [3] [4] [5] . Let a function be analytic in , if there exists a point such that

and with, then

where

And.

Definition 1. A function is said to be in the class if

For some.

Remark

When then is the class studied by [1] .

Definition 2. For functions the Salagean differential operator [6] 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

Proof. Let

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

therefore,

If a point which satisfies and

then by lemma [2]

and

Now,

Since,

But

Let

then

Hence,.

It implies that

is a minimum

point of.

Therefore, we have that

which contradicts the condition of the theorem.

Hence, it is concluded from lemma [2] that

so that

Acknowledgements

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

ReferencesOwa, S., Nunokawa, M., Saitoh, H., Ikeda, A. and Koike, N. (1997) Some Results for Strongly Functions. Journal of Mathematical Analysis and Applications, 212, 98-106. https://doi.org/10.1006/jmaa.1997.5468Aouf, M., Dziok, J. and Sokol, J. (2011) On a Subclass of Strongly Starlike Functions. Applied Mathematics Letters, 24, 27-32. https://doi.org/10.1016/j.aml.2010.08.004Nunokawa, M. (1992) On Properties of Non-Caratheodory Functions. Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, 68, 152-153. https://doi.org/10.3792/pjaa.68.152Nunokawa, M. (1993) On the Order of Strongly Starlikeness of Strongly Convex Functions. Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, 69, 234-237. https://doi.org/10.3792/pjaa.69.234Obradovic, M. and Owa, S. (1989) A Criterion for Starlikeness. Mathematische Nachrichten, 140, 97-102. https://doi.org/10.1002/mana.19891400109Salagean, G.S. (1983) Subclasses of Univalent Functions. Lecture Notes in Math. Springer-Verlag, Heidelberg and New York, 1013, 362-372.