Meromorphic Functions Partially Share Three Values with Their Difference Operators

We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field K, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman’s alternative theorem both in K and in any open disk. Here the main hypothesis is based on the behaviour of |f|(r) when r tends to +∞ on properties of special values and quasi-exceptional values.We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman’s conjectures for n = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f 2 assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on K admits no infinite stationary subsequence, then either f′ + af 2 has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf 3 and f′ + bf 4 have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.

In the present paper, we will consider the class of meromorphic starlike functions with fixed residue .   Silverman et al. (2008) has obtained sharp upper bounds for Fekete-Szegö like functional  for certain subclasses of meromorphic functions. In this paper, we will find sharp upper bounds for  for the class meromorphic starlike functions with fixed residue . The aim of the present paper, is to completely solve Fekete Szegö problem for a certain subclass of meromorphic starlike functions with fixed residue d.   Key words: Fekete-Szegö inequality, starlike function, analytic function, subordination, meromorphic function.

Let f be a meromorphic function in the unit disc and ( a ν ) ν = 1 k ${(a_\nu )_{\nu =1}^k}$ a set of distinct meromorphic functions small with respect to f. An analogue of the second main theorem for f and ( a ν ) ν = 1 k ${(a_\nu )_{\nu =1}^k}$ is given. Upper limits for the sum of defects of an admissible meromorphic function and an admissible holomorphic function follow. For meromorphic and holomorphic functions in the unit disc and their small functions the analogues of Ullrich's theorem are presented.

<p>In this paper, a necessary and sufficient coefficient are given for functions in a class of complex valued meromorphic harmonic univalent functions of the form f = h + g using Salagean operator. Furthermore, distortion theorems, extreme points, convolution condition and convex combinations for this family of meromorphic harmonic functions are obtained.</p>

The function $ \xi(z)$ is obtained from the logarithmic derivative function $\sigma(z)$. The elliptic function $ \wp(z) $ is also derived from the $ \xi(z) $ function. The function $ \wp(z) $ is a function of double periodic and meromorphic function on lattices region. The function $ \wp(z) $ is also double function. The function $ \varphi(z) $ meromorphic and univalent function was obtained by the serial expansion of the function $ \wp(z)$. The function $ \varphi(z) $ obtained here is shown to be a convex function.

We investigate the uniqueness of transcendental analytic fun- ctions that share three values DM in one angular domain instead of the whole complex plane. 1. Introduction and main results In this paper, a transcendental meromorphic (analytic) function is mero- morphic (analytic) in the whole complex plane C and not rational. We as- sume that the reader is familiar with the Nevanlinna's theory of meromorphic functions and the standard notations such as m(r, f ), T (r, f ). For references, see (2). We say that two meromorphic functions f and g share the value a (a ∈ C = C ∪ {∞}) in X ⊆ C provided that in X, we have f (z) = a if and only if g(z) = a. We will state whether a shared value is by DM (differential mul- tiplicities), or by IM (ignoring multiplicities). R. Nevanlinna (see (4)) proved that if two meromorphic functions f and g have five distinct IM shared values in X = C, then f (z) ≡ g(z). After his very work, the uniqueness of meromor- phic functions with shared values in the whole complex plane attracted many investigations (for references, see (7)). E. Mues consider DM shared values and proved the following theorem.

Let $\mathbb D$ be the complex unit disc and let $\Sigma_k$ be the class of all meromorphic functions $f\in\mathbb D\setminus\{0\}$ of the form: $$f(z)=\frac{1}{z}+a_kz^k+a_{k+1}z^{k+1}+\cdots\;k\in\mathbb N,\;a_k\neq 0.$$ A function in $\Sigma:=\Sigma_0$ is called (meromorphic) starlike if $$\mathrm{Re}\left[ -\frac{zf^{\prime }(z)}{f(z)}\right] >0 \text{ in}\;\mathbb D\setminus\{0\}$$ and $\Sigma_k^*$ is the subclass of starlike (meromorphic) functions in $\Sigma_k$. The purpose of the article is to improve many previous results by giving suffcient conditions on the (given) analytic functions (in $\mathbb D$) $g$ and $h$ and on the numbers $k,m\in\mathbb N$ and $c>0$ so that the integral operator $$I_{g,h}^c(f)(z)=\frac{c}{g^{c+1}(z)}\int_0^zf(t)g^c(t)h(t)dt,\;z\in\mathbb D$$ is well-defined in $\Sigma$ and maps $\Sigma_k^*$ into $\Sigma_m^*$. An example that cannot be obtained from the previous results is also provided.

In this study, it is mentioned that meromorphic functions are univalent functions that are analytical everywhere. Complex analytical transformations were investigated by mentioning the necessary form for f (z) to have meromorphic function. It is a function that satisfies the condition   0 hz  . For analytic functions of f and g in the D unit disk, ()fz shows the meromorphic function class with P and subclasses of the P meromorphic analytical function class using the subordination principle between functions with the help of Hadamard product and linear operators. In this way proves is provided.

This article studies the problem of the uniqueness of meromorphic functions that weighted sharing three values which improve some results given by Yi [Theorem 4, Yi, H.X., 1995, Unicity theorems for meromorphic functions that share three values. Kodai Mathematical Journal, 18, 300-314] and Ueda [Ueda, H., 1983, Unicity theorems for meromorphic or entire functions II. Kodai Mathematical Journal, 6, 26-36] and other authors. An application of these new results, if f and g are two distinct nonconstant meromorphic functions sharing 0, 1 and CM, and a is a nonconstant rational function, then N2)(r,1/(g-a))= [image omitted] An example shows that the latter result is not true for some transcendental small functions of f and g.

To consolidate or adapt to many studies on meromorphic functions, we define a new subclass of meromorphic functions of complex order involving a differential operator. The defined function class combines the concept of spiral-like functions with other studies pertaining to subclasses of multivalent meromorphic functions. Inclusion relations, integral representation, geometrical interpretation, coefficient estimates and solution to the Fekete-Szegö problem of the defined classes are the highlights of this present study. Further to keep up with the present direction of research, we extend the study using quantum calculus. Applications of our main results are given as corollaries.

Coefficient estimates for certain classes of meromorphic bi-univalent functions

We study the uniqueness of meromorphic functions concerning nonlinear differential polynomials sharing a nonzero polynomial IM.Though the main concern of the paper is to improve a recent result of the present author [12], as a consequence of the main result we also generalize two recent results of X. M. Li and L. Gao [11].

The theory of entire and meromorphic functions is a very important area of complex analysis. This monograph aims to expand the discussion about some growth properties of integer translated composite entire and meromorphic functions on the basis of their (p,q,t)L -order and (p,q,t)L -type. This book presents six chapters. Chapter 1 introduces the reader to the preliminary definitions and notations. Chapter 2 and Chapter 3 discuss some results related to (p; q; t) L-th order and (p; q; t)L-th lower order of composite entire and meromorphic functions on the basis of their integer translation. Chapter 4 establishes some relations of integer translated composite entire and meromorphic functions based on their (p; q; t) L-th type and (p; q; t) L-th weak type. Chapter 5 deals with some results about (p; q; t) L-th order and (p; q; t) L-th type of composite entire and meromorphic functions on the basis of their integer translation. Chapter 6 focuses on some results about (p; q; t) L-th order and (p; q; t) L-th type of composite entire and meromorphic functions on the basis of their integer translation. This monograph will be very helpful for postgraduates, researchers, and faculty members interested in value distribution theorems in complex mathematical analysis.

Let K be a complete algebraically closed p-adic field of characteristic zero.We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk.Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions.We show that if f P (f ) and g P (g) share a small function α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P satisfy certain inequalities.A breakthrough in this paper consists of replacing inequalities n ≥ k + 2 or n ≥ k + 3 used in previous papers by Hypothesis (G).In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with (q -1) times the characteristic function of the considered meromorphic function.Notation and definition: Let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value | .|.We will denote by E a field that is either K or C. Throughout the paper we denote by a a point in K. Given R ∈]0, +∞[ we define disksThe definition of polynomials of uniqueness was introduced in [19] by P. Li and C. C. Yang and was studied in many papers [11], [13], [20] for complex functions and in [1], [2], [9], [10], [17], [18], for p-adic functions.Throughout the paper we will denote by P (X) a polynomial in E[X] such that P (X) is of the form b l i=1 (X -a i ) ki with l ≥ 2 and k 1 ≥ 2. The polynomial P will be said to satisfy Hypothesis (G) if P (a i ) + P (a j ) = 0 ∀i = j.We will improve the main theorems obtained in [5] and [6] with the help of a new hypothesis denoted by Hypothesis (G) and by thorougly examining the situation with p-adic and complex analytic and meromorphic functions in order to avoid a lot of exclusions.Moreover, we will prove a new theorem completing the 2nd Main Theorem for p-adic meromorphic functions.Thanks to this new theorem we will give more precisions in results on value-sharing problems.

Here we investigate a majorization problem involving starlike meromorphic functions of complex order belonging to a certain subclass of meromorphic univalent functions defined by an integral operator introduced recently by Lashin.