Generalization of Jack’s Lemma for Functions With Fixed Initial Coefficient and Its Applications

Let α e [0,1]p be an analytic function in the unit disc E q and h be analytic and univalent functions in E, Φ be an analytic function in a certain domain D⊃ q (E). The authors study the differential subordinations and where z⊂E Examples of the application of these results are given.

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.

For a fixed analytic univalent function φ, the class of meromorphic univalent α-convex functions with respect to φ is introduced. A representation theorem for functions in the class, as well as a necessary and sufficient condition for functions to belong to the class are obtained. Also we obtain a sharp growth theorem and estimate on a certain coefficient functional for meromorphic starlike functions with respect to φ. Differential subordination and superordination conditions are also obtained for the subclass of meromorphic starlike functions with respect to φ.

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general coefficients | a n | of functions in this subclass are found by using the Faber polynomial expansion. We have thereby generalized and improved some of the previously published results.

We used the concept of quantum calculus (Jackson’s calculus) in a recent note to develop an extended class of multivalent functions on the open unit disk. Convexity and star-likeness properties are obtained by establishing conditions for this class. The most common inequalities of the proposed functions are geometrically investigated. Our approach was influenced by the theory of differential subordination. As a result, we called attention to a few well-known corollaries of our main conclusions.

The goal of this paper is to investigate some of the features of differential subordination of analytic univalent functions in an open unit disc. In addition, it has shed light on geometric features such as coefficient inequality, Hadamard product qualities, and the Komatu integral operator. Some intriguing results for third-order differential subordination and superordination of analytic univalent functions have been installed. Then, using the convolution of two linear operators, certain results of third order differential subordination involving linear operators were reported. As a result, we use features of the Komatu integral operator to analyze and study third-order subordinations and superordinations in relation to the convolution. Finally, several results for third order differential subordination in the open unit disk using generalized hypergeometric function have been addressed using the convolution operator.

Let $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x ( p , q ) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula and The following problem was posed by Paul Gauthier: if f (z) and g (z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f (z) and g (z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D . It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.

Let K be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the p-adic Nevanlinna theory to functional equations of the form \(g = R \circ f\), where R ∈ K(x), f, g are meromorphic functions in K, or in an “open disk”, g satisfying conditions on the order of its zeros and poles. In various cases we show that f and g must be constant when they are meromorphic in all K, or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular, we have an easy way to obtain again Picard-Berkovich’s theorem for curves of genus 1 and 2. These results apply to equations f m + g n = 1, when f, g are meromorphic functions, or entire functions in K or analytic functions in an “open disk”. We finally apply the method to Yoshida’s equation \( {{y'}^{m}} = F(y), \), when F ∈ K(X), and we describe the only case where solutions exist: F must be a polynomial of the form A(y—a) d where m — d divides m, and then the solutions are the functions of the form \(f\left( x \right) = \alpha + \lambda {\left( {x - \alpha } \right)^{\frac{m}{{m - d}}}}\) with \({\lambda ^{m - d}}{\left( {\frac{m}{{m - d}}} \right)^m} = A\).

Ever since Lotfi A. Zadeh published the paper "Fuzzy Sets" in 1965 setting the basis of a new theory named fuzzy sets theory, many scientists have developed this theory and its applications. Mathematicians were especially interested in extending classical mathematical results in the fuzzy context. Such an extension was also done relating fuzzy sets theory and geometric theory of analytic functions. The study begun in 2011 has many interesting published outcomes and the present paper follows the line of the previous research in the field. The aim of the paper is to give some references related to the connections already made between fuzzy sets theory and geometric theory of analytic functions and to present some new results that might prove interesting for mathematicians willing to enlarge their views on certain aspects of the merge between the two theories. Using the notions of fuzzy differential subordination and the classical notion of differential subordination for analytic functions, two criteria for the univalence of the analytic functions are stated in this work.

Background The study of theory for analytic univalent and multivalent functions is an old subject in mathematics, particularly in complex analysis, that has captivated a great deal of scholars owing to the sheer sophistication of its geometrical features as well as its many research possibilities. The study of univalent functions is one of many significant elements of complex analysis for both single and multiple variables. Investigators have become keen on the conventional investigation of this topic since at least 1907. Numerous scholars in the area of complex analysis have emerged since then, including Euler, Gauss, Riemann, Cauchy, and other people. Geometric function theory combines geometry and analysis. Methods This study employs the differential subordination technique to derive multiple characteristics from the new linear operator M σ , μ n , ς Υ ( s ) . The concept of the differential subordination subclass of analytical univalent functions is analyzed. Results In this section, We studied some results on differential subordination and superordination using a specific class of univalent functions stated on a specific space of univalent functions stated on the open unit disc. Using properties of the operator, we discovered a number of properties of superordinations and subordinations related to the idea of the Hadamard product. We investigated several aspects of superordinations and subordinations using a new operator M σ , μ n , ς Υ ( s ) . Conclusions A new operator M σ , μ n , ς Υ ( s ) : Λ ⟶ Λ has been established in this paper connected to the Dziok-Srivastava operator T σ n and the Hadamard product corresponding to the Komatu integral operator Ω μ ς . The difference operator M σ , μ n , ς ϒ ( s ) can have specific properties derived by applying the differential subordination technique. And the objective of this paper is to make use of the connection ( β 1 μ + 1 ) M σ , μ n + 1 , ς ϒ ( s ) = w ( M σ , μ n , ς ϒ ( s ) ) ′ + β 1 μ ( M σ , μ n , ς ϒ ( s ) ) .

The Cauchy integral theorem and the relevant formula (or, equivalently, complex path-independent integrals) have been used in a long series of papers for the determination of zeros and poles of analytic and meromorphic functions. Here this approach is generalized to become applicable to the problem of location of a straight crack inside an infinite plane isotropic elastic medium. The complex path-independent integrals used here contain the first complex potential Φ(z) of Kolosov-Muskhelishvili, which can be obtained experimentally. The present method can be modified to apply to a variety of problems where discontinuity intervals of analytic (or, rather, sectionally analytic) functions are sought.

1. Some problems arising in the study of value distribution of functions of two complex variables. One of the objectives of modern analysis consists in the generalization of methods of the theory of functions of one complex variable in such a way that the procedures in the revised form can be applied in other fields, in particular, in the theory of functions of several complex variables, in the theory of partial differential equations, in differential geometry, etc. In this way one can hope to obtain in time a unified theory of various chapters of analysis. The method of the kernel function is one of the tools of this kind. In particular, this method permits us to develop some chapters of the theory of analytic and meromorphic functions f(z1, * * *, zn) of the class j2(I2n), various chapters in the theory of pseudo-conformal transformations (i.e., of transformations of the domains V2n by n analytic functions of n complex variables) etc. On the other hand, it is of considerable interest to generalize other chapters of the theory of functions of one variable, at first to the case of several complex variables. In particular, the study of value distribution of entire and meromorphic functions represents a topic of great interest. Generalizing the classical results about the zeros of a polynomial, Hadamard and Borel established a connection between the value distribution of a function and its growth. A further step of basic importance has been made by Nevanlinna and Ahlfors, who showed not only that the results of Hadamard and Borel in a sharper form can be obtained by using potential-theoretical and topological methods, but found in this way important new relations, and opened a new field in the modern theory of functions. As one passes from the theory of functions of one variable to the case of two and more variables(2), the question of the value distribution becomes more complex and many directions arise which can be considered as a generalization of the above mentioned chapter of the theory of functions of one variable. However, the formulation of the problems which can be answered, and generalization of methods used in one variable, is not immediate. A function of two complex variables may vanish in a domain on a segment

For which functions $f$ does $A\in\mathbb{G} \Rightarrow f(A)\in\mathbb{G}$ when $\mathbb{G}$ is the matrix automorphism group associated with a bilinear or sesquilinear form? For example, if $A$ is symplectic when is $f(A)$ symplectic? We show that group structure is preserved precisely when $f(A^{-1}) = f(A)^{-1}$ for bilinear forms and when $f(A^{-*}) = f(A)^{-*}$ for sesquilinear forms. Meromorphic functions that satisfy each of these conditions are characterized. Related to structure preservation is the condition $f(\overline{A}) = \overline{f(A)}$, and analytic functions and rational functions satisfying this condition are also characterized. These results enable us to characterize all meromorphic functions that map every $\mathbb{G}$ into itself as the ratio of a polynomial and its ``reversal,'' up to a monomial factor and conjugation. The principal square root is an important example of a function that preserves every automorphism group $\mathbb{G}$. By exploiting the matrix sign function, a new family of coupled iterations for the matrix square root is derived. Some of these iterations preserve every $\mathbb{G}$; all of them are shown, via a novel Frechet derivative-based analysis, to be numerically stable. A rewritten form of Newton's method for the square root of $A\in\mathbb{G}$ is also derived. Unlike the original method, this new form has good numerical stability properties, and we argue that it is the iterative method of choice for computing $A^{1/2}$ when $A\in\mathbb{G}$. Our tools include a formula for the sign of a certain block $2\times 2$ matrix, the generalized polar decomposition along with a wide class of iterations for computing it, and a connection between the generalized polar decomposition of $I+A$ and the square root of $A\in\mathbb{G}$.

In this paper we consider the meromorphic function G(z) with a pole of order 1 at -a and analytic function F(z) with a zero -a of order 2 in <TEX>$\mathbb{D}=\{z :{\mid}z{\mid}</TEX><TEX><</TEX><TEX>1\}$</TEX>, where -1 < a < 1. From these functions we obtain the regular simply-connected minimal surface <TEX>$S=\{(u(z),\;{\nu}(z),\;H(z)):z{\in}\mathbb{D}\}$</TEX> in <TEX>$E^3$</TEX> and the harmonic function <TEX>$f=u+i{\nu}$</TEX> defined on <TEX>$\mathbb{D}$</TEX>, and then we investigate properties of the minimal surface S and the harmonic function f.