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New Result for Strongly Starlike Functions

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1. Introduction

Let be the class of functions of the form

(1)

which are analytic in the unit disk. A function is said to be starlike of order if and only if

(2)

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

(3)

We denote by the subclass of consisting of functions which are convex of order in.

If satisfies

(4)

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

If satisfies

(5)

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

(6)

where

And.

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

(7)

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

(8)

Taking the logarithmic differentiation in both sides of Equation (8), we have

(9)

Multiply Equation (9) through by, to get

(10)

Multiply Equation (10) by to obtain

(11)

Multiply Equation (11) through by 2 and divide through by to give

(12)

Multiplying Equation (12) by, and further simplifica-

tion, we obtain

(13)

therefore,

(14)

If a point which satisfies and

then by lemma [2]

and

Now,

(15)

Since,

(16)

(17)

But

Let

then

(18)

Hence,.

It implies that

is a minimum

point of.

Therefore, we have that

(19)

which contradicts the condition of the theorem.

Hence, it is concluded from lemma [2] that

(20)

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

*Applied Mathematics*,

**8**, 324-328. doi: 10.4236/am.2017.83027.

[1] |
Owa, 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.1997.5468 |

[2] |
Aouf, 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.2010.08.004 |

[3] |
Nunokawa, 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.68.152 |

[4] |
Nunokawa, 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.69.234 |

[5] |
Obradovic, M. and Owa, S. (1989) A Criterion for Starlikeness. Mathematische Nachrichten, 140, 97-102. https://doi.org/10.1002/mana.19891400109 |

[6] | Salagean, G.S. (1983) Subclasses of Univalent Functions. Lecture Notes in Math. Springer-Verlag, Heidelberg and New York, 1013, 362-372. |

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