FEKETE-SZEGÖ INEQUALITIES FOR QUASI-SUBORDINATION FUNCTION USING NEW SUBCLASS

A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to H", the linear invariant family of locally univalent functions of finite order or.If f(z) is subordinate to F(z) in the open unit disc, /'(0) > 0, and F(z) is in U" 1.65 < a < oo, then f'(z) is majorized by F'(z) in \z\ < (a + 1) -(a2 + 2a)vi.The result is sharp.I. Introduction.Let <S denote the set of all normalized analytic univalent functions in the open unit disc D. Let/(z), F(z) and <p(z) be analytic in |z| < r.We say that/(z) is majorized byLet tla be the set of all locally univalent (f'(z) = 0) analytic functions in D with order < a which are of the form f(z) = z A-.The family U" is known as the universal linear invariant family of order a [4].A concise summary and introduction to properties of linear invariant families which relate to the following material is contained in [1].The present paper concludes the proof of results announced in [1].Majorization-subordination theory begins with Biernacki who showed in 1936 that if/(z) is subordinate in D to F(z) (F(z) G <S), then/(z) is majorized by F(z) in |z| < 1/4.In the succeeding years Goluzin, Tao Shah, Lewandowski and MacGregor examined various related problems but always under the stipulation that the dominant function F(z) is in <S (for greater detail see [1]).In 1951 Goluzin showed that if f(z) is majorized by a univalent function F(z), then f'(z) would be majorized by F'(z) in |z| < 0.12.He conjectured that majorization would always occur for jz| < 3 -y and this was proved by Tao Shah in 1958.In this paper we show that the result is actually true for functions in 11. and obtain the sharp radius of majorization as a + 1 -(a2 + 2a)^2 for 1.65 < o < oo.This yields 3 -y for the case a = 2.Our investigation shows that the important datum for majorization-subordination theory is not univalence, but the order of a linear invariant family.In particular, many classically derived estimates for univalent functions are true for functions of infinite valence.

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.

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 ) ) .

A normalized analytic function f defined on the open unit disc D is called Ma-Minda convex if 1 + zf′′(z)/f′(z) is subordinate to the function φ. For 0 ⩽ α ⩽ β, the Kaplan class K(α, β) of type α and β consists of normalized analytic functions of the form p^{α}g defined on D where p with p(0) = 1 is an analytic function taking values in the right half-plane and g is an analytic function with g(0) = 1 satisfying Re(zg′(z)/g(z)) &gt; (α − β)/2. For functions f with f′ ∈ K(α, β), we obtain the radius of Ma-Minda convexity for various choices of φ. The radius of lemniscate convexity, lune convexity, nephroid convexity, exponential convexity and several other radius estimates are examined. The results obtained are sharp.

Let T n ( A , B , α ) denote the class of functions of the form: f ( z ) = z + ∑ k = n + 1 ∞ a k z k ( n ∈ N = { 1 , 2 , 3 , … } ) , which are analytic in the open unit disk U and satisfy the following subordination condition: f ′ ( z ) + α zf ″ ( z ) ≺ 1 + Az 1 + Bz ( z ∈ U ; - 1 ⩽ B < 1 ; B < A ; α > 0 ) . In this paper, we find the sharp lower bound on | z | = r < 1 for the functional Re { f ′ ( z ) + α zf ″ ( z ) } over the class: T n ( A , B , 0 ) = f ( z ) = z + ∑ k = n + 1 ∞ a k z k : f ′ ( z ) ≺ 1 + Az 1 + Bz ( n ∈ N = { 1 , 2 , 3 , … } ; z ∈ U ) . By applying this result, several interesting consequences are given.

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.

Introduction. In his study of the structure of distribution kernels (centering around his celebrated kernel theorem), L. Schwartz [11; 13] has studied the subclasses of regular and very regular distribution kernels, which may be characterized roughly by the fact that they carry infinitely differentiable functions into infinitely differentiable functions. It is our object in the present paper to extend this study to kernels which are analytically regular, i.e., which, again roughly, carry analytic functions into analytic functions. The motivation for such a study is provided by the fundamental solutions of elliptic equations with analytic coefficients. One consequence of our results is that for such equations the standard theorems on the analyticity of regular solutions imply the analyticity of the fundamental solution. The principal tool in our investigation is the general theory of topological vector spaces (particularly in certain forms given to it by A. Grothendieck) and especially the theory of topological tensor products. In ?1.1, we summarize the definitions from the theory of topological vector spaces which we shall use. In ?1.2, we give a brief summary of the definitions and principal results of Grothendieck'stheory of topological tensor products. In ?I.3, we summarize the results of Schwartz's study of distribution kernels, at least insofar as they relate to the generalizations to be given to analytic kernels. ?II is devoted to an intensive consideration of the space Gt(K) of real analytic functions on a compact subset of Rn. ?III states and proves our principal results on analytic distribution kernels. The writer should like to thank Professors L. Nachbin and F. E. Browder for discussion, suggestions, and criticism. The results of the present paper constitute a portion of the writer's Doctoral Dissertation [1] at the University of Sao Paulo, prepared while in residence at the Institute for Pure and Applied Mathematics, Rio de Janeiro.

Neighborhoods of certain analytic functions

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on D. We obtain as corollaries of this purely function-theoretic work, new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as S^p and the weighted Hardy spaces H^2(beta).

Let Ω denote the class of functions f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ belonging to the normalized analytic function class A in the open unit disk U = z : z &lt; 1 , which are bi-univalent in U , that is, both the function f and its inverse f − 1 are univalent in U . In this paper, we introduce and investigate two new subclasses of the function class Ω of bi-univalent functions defined in the open unit disc U , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

It is well known that the conformable and the symmetric differential operators have formulas in terms of the first derivative. In this document, we combine the two definitions to get the symmetric conformable derivative operator (SCDO). The purpose of this effort is to provide a study of SCDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to impose two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SCDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function and convolution structures. Moreover, by using the SCDO, we introduce a generalized class of Briot–Bouquet differential equations to introduce, what is called the symmetric conformable Briot–Bouquet differential equations. We shall show that the upper bound of this class is symmetric in the open unit disk.

In Geometric Function Theory, it is well known that the familiar Koebe function f(z) = z/(1 − z)2 is the extremal function for the class 𝒮* of starlike functions in the open unit disk 𝕌 and also that the function g(z) = z/(1 − z) is the extremal function for the class 𝒦 of convex functions in the open unit disk 𝕌. However, the partial sum fn (z) of f(z) is not starlike in 𝕌 and the partial sum gn (z) of g(z) is not convex in 𝕌. The aim of the present paper is to investigate the starlikeness and convexity of these partial sums fn (z) and gn (z). Computational and graphical usages of Mathematica (Version 4.0) as well as geometrical descriptions of the image domains in several illustrative examples are also presented. E-mail: owa@math.kindai.ac.jp

This study of Hankel and Hermitian Toeplitz determinants is one of the major areas of interest in Geometric function theory and has wide applications in the areas of signal processing and Applied Mathematics. In our present investigations, we define a new subclass of normalized analytic functions H(λ)(λ≥0), defined using a subordination relation with the sine function K(z)=1+sinz. For the class H(λ), coefficient estimates, upper and lower bounds for the Hermitian Toeplitz determinants of second and third order are found. In addition, estimates are provided for the second and third-order Hankel determinants for the class H(λ).

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.

Let the differential system where D is the diagonal matrix and the elements αij (Z) of the matrix A(z) are analytic functions in some neighborhood of the closed unit disc. In this paper under several assumptions with respect to the constant matrices and the diagonal D, it is proved that the conjugate system of (1) has exactly k(p − l) linearly independent solutions in the product space: where is the usual Hilbert space of analytic functions in the open unit disc.