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Research Article

Open Access Peer-reviewed

Ajai P.T.^{ }, Moses B.O., Ihedioha S.A.

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

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.

Let denote the class of functions of the form

(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

(1.2) |

**Definition 2**

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

(1.3) |

**Definition 3**

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

(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

(1.5) |

Define as

(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;

(1.7) |

Using subordination we write (1.7) as

(1.8) |

Where is as defined in (1.6)

**Theorem 2.1**

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

(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

(2.2) |

Which is equivalent to

Notice that since we similarly have

(2.3) |

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

(2.4) |

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

(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.

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

(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

(3.2) |

and . Then

**Proof**

It is obvious that

Using theorem 3.1 we have to show that

(3.4) |

Suppose we set. Then we have

Notice that and from theorem 2.1 we can write

This proves our theorem.

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

(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.

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.

[1] | Gerhardt C. I., Leibniz G.W, Mathematische Schriften III/1, Georg Olms, NY, USA, 1971. | ||

In article | |||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[4] | Bajpai S.K, A note on a class of meromorphic univalent functions, Rev. Rownanie Math. Pures Appl. 22, 295-2971977. | ||

In article | |||

[5] | Goncharov A. B, Polylogarithms in arithmetic and geometry, Proceedings of the International Congress of Mathematicians, 374-387, Zurich, Switzerland, August 1994. | ||

In article | View Article | ||

[6] | Goel R.M. and S o h i N. S, on a class of meromorphic functions, Glasnik Matematioki 17 (1981), 19-28. | ||

In article | |||

[7] | Rashhed K. A, Darus M., A new class of meromorphicfunctions involving polylogarithm function: Journal of complex analysis, (2014), 135-140. | ||

In article | View Article | ||

[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). | ||

In article | View Article | ||

[9] | Littlewood J.E, On inequalities in the theory of functions, Proceedings of the London Mathematical Society, 23(1) 481-519, 1925. | ||

In article | View Article | ||

[10] | Robertson M.S, Convolutions of schlicht functions, Proc. Amer. Math. Soc. 13 (.1962), 585-589. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2019 Ajai P.T., Moses B.O. and Ihedioha S.A.

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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*. Vol. 7, No. 5, 2019, pp 167-170. http://pubs.sciepub.com/ajams/7/5/2

P.T., Ajai, Moses B.O., and 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 (2019): 167-170.

P.T., A. , B.O., M. , & S.A., I. (2019). 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.

P.T., Ajai, Moses B.O., and 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, no. 5 (2019): 167-170.

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[1] | Gerhardt C. I., Leibniz G.W, Mathematische Schriften III/1, Georg Olms, NY, USA, 1971. | ||

In article | |||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[4] | Bajpai S.K, A note on a class of meromorphic univalent functions, Rev. Rownanie Math. Pures Appl. 22, 295-2971977. | ||

In article | |||

[5] | Goncharov A. B, Polylogarithms in arithmetic and geometry, Proceedings of the International Congress of Mathematicians, 374-387, Zurich, Switzerland, August 1994. | ||

In article | View Article | ||

[6] | Goel R.M. and S o h i N. S, on a class of meromorphic functions, Glasnik Matematioki 17 (1981), 19-28. | ||

In article | |||

[7] | Rashhed K. A, Darus M., A new class of meromorphicfunctions involving polylogarithm function: Journal of complex analysis, (2014), 135-140. | ||

In article | View Article | ||

[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). | ||

In article | View Article | ||

[9] | Littlewood J.E, On inequalities in the theory of functions, Proceedings of the London Mathematical Society, 23(1) 481-519, 1925. | ||

In article | View Article | ||

[10] | Robertson M.S, Convolutions of schlicht functions, Proc. Amer. Math. Soc. 13 (.1962), 585-589. | ||

In article | View Article | ||