Certain new subclasses of multivalent meromorphic function

Introduction

The Riemann mapping theorem in complex analysis gave a new idea to study the complex functions. In which the geometry of complex functions is studied and this new branch called Geometric Function Theory. Latter in 1907 P. Koebe initiate works for univalent function. Other geometric properties like starlike function, convex function, close-to-convex function, spiral like function and multivalent function were defined and studied initially by Robertson, Lowner, Kaplan, Spacek, Libera. Latter many other authors contributed their works in the field of geometric function theory. In the analytic function theory, there have been many differential implications in which characterization of the functions is determined by a differential condition. If f (z) is analytic in a convex domain D and   0

Most of the special classes of functions for which Bieberbach's conjecture is known to hold depends on the image domain having a specific geometric characterization which can then be expressed in the analytic form. One of the earliest results in this direction concerns the star like functions. A single valued function f (z) is said to be univalent or Schlicht in a domain D  C (the complex plane), if it never takes on the same values twice, that is, if f (z1) = f (z2) for z1, z2  D implies that z1 = z2. In the other word F is one to one on D. In the case the equation f (z) = w has at most one root in D for any complex number w. Let A denote the class of function } > 0, (𝑧 ∈ 𝑈).

The set E is said to be convex, if the line segment joining any two points of E lies entirely in E. This is equivalent to saying that tz1 + (1-t) z2  E for every z1, z2  E, 0  t < 1. A function in S is said to be convex of order () (0   < 1), if any only if

𝑓 ′ (𝑧) } > 𝛼, (𝑧 ∈ 𝑈). K () denotes such class of convex functions. It is observed that K ()  K (0)  K, where K being the class of convex functions with respect to origin in U. This means that f  K, if any only if ℜ𝑒 {1 + 𝑧𝑓 ″ (𝑧) 𝑓 ′ (𝑧) } > 0, (𝑧 ∈ 𝑈).

Some Related Definitions Definition 2.1 A function

} > 0, (𝑧 ∈ 𝑈), and the class of all such functions is denoted by C. Using Noshiro-Warschawski result, it can be proved that every close-to-convex function is univalent. It is easy to see from the definition that every convex and starlike function is close-to-convex. Thus, we infer that 𝜅 ⊂ 𝑆 * ⊂ 𝐶 ⊂ 𝑆.

Definition 2.2 A function

is said to be close-to-convex of order𝛼 (0 < 𝛼 < 1), if there exists a convex function g

). We denote the class of all such -spirallike functions by 𝑆𝑝(). It was also shown by Spacek that -spirallike functions are always univalent in the unit disk U.

Definitions 2.4 A function

). We denote the class of all such type of functions by Sp (, ) which was introduced by Libera [18].

Definitions 2.5 A function

). We denote the class of all such functions by M (, ) which were studied by Robertson.

When  = 0, then 𝑀(0, 𝛼) = 𝜅(𝛼).

Main Results

Let  m+1 be the class of function f of the form

. If f and g are analytic in E = E  {0}, we say that f is subordinate to g,



For p = 1, the integral operator reduces into the following operator:

Now in this article, we introduce some new class of Meromorphic function using the operator ( )

. To establish our main results we need the following Lemma. Lemma 3.1 [10,11]: Let the function h (z) be analytic and convex (univalent) in E with h (0) = 1, suppose also that the function  (z) given by

Multiplying by z and using recurrence relation, we have

Equation ( 1) and (2) gives us

. Now Differentiating with respect to z, we get

Multiplying by z and using recurrence relation, we have.

Conclusion

In the research work, we will discuss some new classes of analytic functions using the technique of convolution, certain liner operators will be defined all classes of analytic function. The main object of the present paper is to introduce the certain subclasses of Meromorphic multivalent functions involving. We discuss some interesting geometric properties according to coefficient inequality.

Future Scope

There are several other operators which can be applied on multivalent meromorphic functions. The theorems which are derived in this research article can be applied practically for solving some functional analysis theoretical problems.

Acknowledgement

Acknowledgements:The authors thank Shri Radheshyam R. Morarka Government College, Nawalgarh (Jhunjhunu) and Seth GB Podar College, Nawalgarh, for their support and resources that made this research possible.

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