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Properties of a Certain Class of Meromorphic Analytic Functions Defined by a Linear Operator

Ajai P.T.*, Moses B.O., Ihedioha S.A.

Department of Mathematics, University of The Gambia, Banjul, The Gambia

Abstract: In this present paper, we introduced and characterized a new class of meromorphic univalent functions associated with polylogarithm by investigating; coefficient inequality, convolutions property, integral means and other properties of the class.

Keywords: analytic function, differential operator, Hadamard product, univalent functions

Received September 10, 2019; Revised October 12, 2019; Accepted October 25, 2019

Citation: Ajai P.T., Moses B.O., Ihedioha S.A., Properties of a Certain Class of Meromorphic Analytic Functions Defined by a Linear Operator. American Journal of Applied Mathematics and Statistics, 7 (5). 167-170. January 2019. doi:10.12691/ajams-7-5-2

1. Introduction and Definitions

Let denote the class of functions of the form

[picture] | (1.1)

Which are analytic in the unit disk . Having a simple pole at the origin with residue 1. Furthermore, let and, denotes the subclasses of which are univalent, meromorphically starlike and convex respectively.

Definition 1

Analytically, a function of the form (1.1) is in if and only if

[picture] | (1.2)

Definition 2

Similarly, If and only if is of the form (1.1) and satisfies

[picture] | (1.3)

Definition 3

For , the set of natural numbers with an absolutely convergent series defined as

[picture] | (1.4)

Is known as the polylogarithm. This class of functions was invented by Liebniz and Bernouli [1]. For more works on polylogarithm and meromorphic functions see [2,3,4,5,6,7].

We state here a linear operator derived as follow;

Let which is defined by the following Hadamard product by Where

[picture] | (1.5)

Define as

[picture] | (1.6)

Definition 4

Let be defined as in (1.1) and as stated in (1.6) then the function then the function in (1.1) is said to be in class if the following geometric condition are satisfy;

[picture] | (1.7)

Using subordination we write (1.7) as

[picture] | (1.8)

Where is as defined in (1.6)

2. Coefficient Inequality

Theorem 2.1

Let of the form (1.1) a function is said to be in the class iff the following bound is satisfy:

[picture] | (2.1)

Proof

Assume that (2.1) holds true then from (1.8) we have

Proving (2.1) Conversely, suppose

We have to show that condition (2.1) is true. Thus we have

[picture] | (2.2)

Which is equivalent to

Notice that since we similarly have

[picture] | (2.3)

We choose the value z on the real axis and letting, we have

[picture] | (2.4)

Which proves our assertion. The result is sharp here for the function;

[picture] | (2.5)

Theorem 2.2

The class is closed under convex combination.

Let then for, then we have.

Proof

By hypothesis and

Then

Thus we have from (2.1) the following

This complete our proof.

3. Integral Means Inequalities

Let and be analytic in U, is said to be subordinate to written as

[picture] | (3.1)

If there exists a Schwarz function which is analytic in U with, such that Furthermore, if the function g(z) is univalent in U, then we have the following equivalence , see [8] .

Theorem 3.1 [9]

If f(g) and g(z) are analytic in U with , then for , and , . Then

Theorem 3.2

Let and be defined by if there exists w(z) such that

[picture] | (3.2)

and . Then

Proof

It is obvious that

Using theorem 3.1 we have to show that

[picture] | (3.4)

Suppose we set. Then we have

Notice that and from theorem 2.1 we can write

This proves our theorem.

4. Convolution Property

Let , and Robbinson [10] has shown that is also in .

Theorem 4.1

Suppose then the Hadamard product or convolution of the functions f and g belongs to the class . Where.

Proof.

Since, from theorem 2.1 we have and

We need to find the largest , by Cauchy-Schwarz inequality, we have

[picture] | (3.5)

Thus it suffices to show that

Which is equivalent to

But from (3.5) we have

The above simplify to. This proves our result.

Acknowledgements

The authors are thankful to the referees for their valuable suggestions. The first Author appreciates the directorate of Technical Aids Corps (TAC) for the privilege accorded me to be deployed as volunteers to The Gambia.

References

[1]   | Gerhardt C. I., Leibniz G.W, Mathematische Schriften III/1, Georg Olms, NY, USA, 1971.
[2]   | Al-Shaqsi K. and Darus M , A multiplier transformation defined by convolution involving nth order polylogarithm functions, International Mathematical Forum, 4 (37), 1823-1837, 2009.
[3]   | Al-Amiri H.S and Reade M.O, On linear combination of some expression in the theory of Univalent functions, Monatsh maths 80, 257-264, 1975.
[4]   | Bajpai S.K, A note on a class of meromorphic univalent functions, Rev. Rownanie Math. Pures Appl. 22, 295-2971977.
[5]   | Goncharov A. B, Polylogarithms in arithmetic and geometry, Proceedings of the International Congress of Mathematicians, 374-387, Zurich, Switzerland, August 1994.
[6]   | Goel R.M. and S o h i N. S, on a class of meromorphic functions, Glasnik Matematioki 17 (1981), 19-28.
[7]   | Rashhed K. A, Darus M., A new class of meromorphicfunctions involving polylogarithm function: Journal of complex analysis, (2014), 135-140.
[8]   | Miller S.S. Mocanu, P.T. Differential Subordinations.Theory and Applications,Series on Monographs and Textbooks in Pure and Appl. Math. No. 255, Marcel Dekker Inc.,New York, (2000).
[9]   | Littlewood J.E, On inequalities in the theory of functions, Proceedings of the London Mathematical Society, 23(1) 481-519, 1925.
[10]   | Robertson M.S, Convolutions of schlicht functions, Proc. Amer. Math. Soc. 13 (.1962), 585-589.