ON A CLASS OF MEROMORPHIC p-VALENT STARLIKE FUNCTIONS INVOLVING CERTAIN LINEAR OPERATORS

Let ∑ p be the class of functions f(z) which are analytic in the punctured disk E∗ = {z ∈ C : 0 < |z| < 1}. Applying the linear operator Dn+p defined by using the convolutions, the subclass n+p(α) of ∑ p is considered. The object of the present paper is to prove that n+p(α) ⊂ n+p−1(α). Since 0(α) is the class of meromorphic p-valent starlike functions of order α, all functions in n+p−1(α) are meromorphic p-valent starlike in the open unit disk E. Further properties preserving integrals and convolution conditions are also considered.

which are analytic in the punctured disk E * = {z ∈ C : 0 < |z| < 1}.The convolution of two power series f (z), given by (1.1) and is defined as the following power series: for some α (0 α < 1).A function f (z) in * p (α) is called a meromorphic p-valent starlike of order α in E.
A function f (z) ∈ p is said to be in the class -n+p−1 (α) if it satisfies the inequality where n is any integer greater than −p, 0 α < 1, and (1.6) The operator D n+p−1 when p = 1 was first introduced by Ganigi and Uralegaddi [1] and then generalized by Yang [9].In recent years, many authors (e.g., [8,10,11]) have investigated certain subclasses of meromorphic functions defined by the operator D n+p−1 .In this paper, we show that a function f (z) ∈ -n+p−1 (α) is meromorphic p-valent starlike of order α.More precisely, it is proved that the starlikeness of members of -n+p−1 (α) is a consequence of (1.7).Further, integral transforms of functions in -n+p−1 (α) and convolution conditions are also considered.

Properties of the class -n+p−1 (α).
In proving our main results, we will need the following lemma.
We have to show that (2.1) implies the inequality Consider the analytic function w(z) in E defined by It is clear that w(0) = 0. Equation (2.3) may be written as Differentiating (2.4) logarithmically and using the identity (easy to verify) we obtain (2.6) We claim that |w(z)| < 1 in E. For otherwise, by Lemma 2.1, there exists a point z 0 in which contradicts (2.1).Hence, Then, Proof.From the definition of F(z), we have (2.10) Using (2.5) and (2.10), condition (2.8) may be written as (2.11) We have to prove that (2.11) implies the inequality Clearly, w(z) is analytic and w(0) = 0. Equation (2.13) may be written as Differentiating (2.14) logarithmically and using (2.5), we obtain (2.15) Using (2.14) and (2.15), we get . (2.16) The remaining part of the proof is similar to that of Theorem 2.2.
According to Theorem 2.2, we have the following corollary at once.
To prove Theorem 2.9, we need the following lemmas.
Theorem 2.9.Let f (z) ∈ -n+p−1 (α) and let β be a complex number with β ≠ 0 and satisfy either and q(z) is the best dominant.
where β is a real number and Proof.From Theorem 2.9, we have Re where w(z) is analytic in E, w(0) = 0, and |w(z)| < 1 for z ∈ E.
To see that the bound 2 2pβ(1−α) cannot be increased, we consider the function f (z) The proof of the corollary is complete.
3. Convolution conditions.In [7], Silverman, Silvia, and Telage considered some convolution conditions for starlikeness of analytic functions.Recently, Silverman and Silvia [6] showed many necessary and sufficient conditions in terms of convolution operators for an analytic function to be in classes of starlike and convex.In this section, we give some necessary and sufficient conditions in terms of convolution operators for meromorphic functions to be in * p (α) and -n+p−1 (α).
which is equivalent to This simplifies to (3.5) Therefore, (3.4) is equivalent to that is, This proves Lemma 3.1.
, an application of (1.6) to Lemma 3.1 yields where g(z) = 1/z p (1 − z) n+p .In view of (3.5), we may write This completes the proof of the theorem.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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1 .
Introduction.Let p denote the class of functions of the form f (z) denote the class of functions of the form (1.1), which satisfy the condition Re zf (z) f (z) < −pα z ∈ E = z ∈ C : |z| < 1 (1.4)