Some classes involving a convolution of analytic functions with some univalency conditions

Introduction to mathematical physics: William Band, Princeton, N.J., Toronto, London, New York, D. Van Nostrand Co., Inc., 1959, X, 326 pp., ill.

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.

In Amer and Darus (Missouri J Math Sci, to appear; Int J Math Anal 6(12):591–597, 2012), the author introduced and studied a linear operator defined on the class of normalized analytic function in the unit disk. This operator is motivated by many researchers. With this operator sharp bound for the nonlinear functional for the class of analytic functions in the open unit disk is obtained. In this paper we discuss sharp bound for the nonlinear functional for the class of analytic functions defined by a linear operator has been considered. Several other results are also considered. There are interesting properties of normalized function in the unit disk for sharp sconced hankel for linear operator. In addition, various other known results are also pointed out. We also find some interesting corollaries on the class of normalized analytic functions in the open unit disk. Our results certainly generalized several results obtained earlier. Therefore, many interesting results could be obtained and we also derive some interesting corollaries of this class. The operator defined can be extended and can solve many new results and properties.

In this paper, we first introduce a linear integral operator ℑp(a,c,μ)(μ>0;a,c∈R;c>a>−μp;p∈N+:={1,2,3,…}), which is somewhat related to a rather specialized form of the Riemann–Liouville fractional integral operator and its varied form known as the Erdélyi–Kober fractional integral operator. We then derive some differential subordination and differential superordination results for analytic and multivalent functions in the open unit disk U, which are associated with the above-mentioned linear integral operator ℑp(a,c,μ). The results presented here are obtained by investigating appropriate classes of admissible functions. We also obtain some Sandwich-type results.

Possibility of Approximation: 1. Basic notions 2. Linear operators 3. Approximation theorems 4. The theorem of Stone 5. Notes Polynomials of Best Approximation: 1. Existence of polynomials of best approximation 2. Characterization of polynomials of best approximation 3. Applications of convexity 4. Chebyshev systems 5. Uniqueness of polynomials of best approximation 6. Chebyshev's theorem 7. Chebyshev polynomials 8. Approximation of some complex functions 9. Notes Properties of Polynomials and Moduli of Continuity: 1. Interpolation 2. Inequalities of Bernstein 3. The inequality of Markov 4. Growth of polynomials in the complex plane 5. Moduli of continuity 6. Moduli of smoothness 7. Classes of functions 8. Notes The Degree of Approximation by Trigonometric Polynomials: 1. Generalities 2. The theorem of Jackson 3. The degree of approximation of differentiable functions 4. Inverse theorems 5. Differentiable functions 6. Notes The Degree of Approximation by Algebraic Polynomials: 1. Preliminaries 2. The approximation theorems 3. Inequalities for the derivatives of polynomials 4. Inverse theorems 5. Approximation of analytic functions 6. Notes Approximation by Rational Functions. Functions of Several Variables: 1. Degree of rational approximation 2. Inverse theorems 3. Periodic functions of several variables 4. Approximation by algebraic polynomials 5. Notes Approximation by Linear Polynomial Operators: 1. Sums of de la Vallee-Poussin. Positive operators 2. The principle of uniform boundedness 3. Operators that preserve trigonometric polynomials 4. Trigonometric saturation classes 5. The saturation class of the Bernstein polynomials 6. Notes Approximation of Classes of Functions: 1. Introduction 2. Approximation in the space 3. The degree of approximation of the classes 4. Distance matrices 5. Approximation of the classes 6. Arbitrary moduli of continuity Approximation by operators 7. Analytic functions 8. Notes Widths: 1. Definitions and basic properties 2. Sets of continuous and differentiable functions 3. Widths of balls 4. Applications of theorem 2 5. Differential operators 6. Widths of the sets 7. Notes Entropy: 1. Entropy and capacity 2. Sets of continuous and differentiable functions 3. Entropy of classes of analytic functions 4. More general sets of analytic functions 5. Relations between entropy and widths 6. Notes Representation of Functions of Several Variables by Functions of One Variable: 1. The Theorem of Kolmogorov 2. The fundamental lemma 3. The completion of the proof 4. Functions not representable by superpositions 5. Notes Bibliography Index.

Problem statement: We introduced a new bijective convolution linear operator defined on the class of normalized analytic functions. This operator was motivated by many researchers namely Srivastava, Owa, Ruscheweyh and many others. The operator was essential to obtain new classes of analytic functions. Approach: Simple technique of Ruscheweyh was used in our preliminary approach to create new bijective convolution linear operator. The preliminary concept of Hadamard products was mentioned and the concept of subordination was given to give sharp proofs for certain sufficient conditions of the linear operator aforementioned. In fact, the subordinating factor sequence was used to derive different types of subordination results. Results: Having the linear operator, subordination theorems were established by using standard concept of subordination. The results reduced to well-known results studied by various researchers. Coefficient bounds and inclusion properties, growth and closure theorems for some subclasses were also obtained. Conclusion: Therefore, many interesting results could be obtained and some applications could be gathered.

Some classes of analytic and multivalent functions involving a linear operator

Hadamard type operators on spaces of real analytic functions in several variables

Motivated by the success of the familiar Dziok–Srivastava convolution operator, we introduce here a closely-related linear operator for analytic functions with fractional powers. By means of this linear operator, we then define and investigate a class of analytic functions. Finally, we determine certain conditions under which the partial sums of the linear operator of bounded turning are also of bounded turning. We also illustrate an application of a fractional integral operator.

This is a paper version of my presentation at Winter School in Complex Analysis and Operator Theory, Valencia February 2010. A real analytic function (i.e., possibly complex valued analytic function of a real argument) is one of the most classical objects of analysis. The theory of the whole class of this functions (treated as a topological vector space) and linear operators on them has developed recently due to new functional analytic tools available. The aim of this course is to survey this development with special emphasis on composition, differential and convolution operators on the space of real analytic functions and to show how our knowledge on the space helps to understand these operators. A nice feature of the theory is that a lot of deep classical theorems of real and complex analysis will be relevant and they will find unexpected relations to functional analytic results. The course will consists of four lectures: 1. Operator relevant properties of the space of real analytic functions • Topology on A (Ω) and tools for study operators used in the course • Relation with the Cousin problem 2. Composition operators on the space of real analytic functions • The space of real analytic functions as an algebra • When it has a closed range, when it is a topological embedding • Relation with analytic/algebraic geometry • How little do we know about hypercyclicity? 3. Differential and convolution operators on the space of real analytic functions • Surjectivity • Relation with algebraic geometry, Fourier analysis and the additive Cousin problem • How little do we know about parameter dependence and solution operators of differential and convolution equations on A (Ω)? 4. Isomorphism of the spaces of real analytic functions • Isomorphic classification for spaces over compact manifolds • Relation with composition and convolution operators • How little do we know about isomorphic classification over non-compact manifolds? We explain main ideas behind the proofs of the results and provide plenty of open problems together with their motivation and background. We try to be reasonably self-contained to make lectures accessible to non-specialists and especially to young mathematicians entering the subject. We consider spaces of real analytic functions over real analytic manifolds (both compact and non-compact). 12000 Mathematics Subject Classification. Primary: 46E10, 46E25, 26E05. Secondary: 14P15, 31A99, 32C05, 32U05, 34A35, 34K06, 35B35, 35E99, 35R10, 44A35, 46A04, 46A13, 46A35, 46A63, 46F15, 46M18, 47A16, 47A80, 47B33.

The main object of the present paper is to derive some results for multivalent analytic functions defined by a linear operator. Making use of a certain operator, which is defined here by means of Hadamard product, we introduce a subclasses \(S_{A,B}^{p,\gamma}(\alpha,\lambda,\mu,\nu,a,c)\) of the class \(A(p)\) of normalized p-valent analytic functions on the open unit disk. Also we have extended some of the previous results and have given necessary and sufficient condition for this class.

Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).

Let q1, q2 be univalent in Δ:= {z: |z| < 1} and p be certain analytic function. We give some applications of first order differential subordinations and superordinations to obtain sufficient conditions to satisfy the following sandwich implication which is a generalization for various known sandwich theorems: $$ \beta zq_1^k (z)q'_1 (z) + \sum\limits_{j = 0}^n {\alpha _j q_1^j (z)} \prec \beta zp^k (z)p'(z) + \sum\limits_{j = 0}^n {\alpha _j p^j (z)} \prec \beta zq_2^k (z)q'_2 (z) + \sum\limits_{j = 0}^n {\alpha _j q_2^j (z)} $$ implies q1(z) ≺ p(z) ≺ q2(z), where k ∈ ℤ and β ≠ 0, α′j ∈ ℂ. Some of its special cases and its applications will be considered for certain analytic functions and certain linear operators.

This paper is concerned mainly with the linear operators \(I_f^{\gamma , \alpha}\) and \(J_f^{\gamma , \alpha}\) of analytic function \(f\).The norm of \(I_f^{\gamma , \alpha}\) and \(J_f^{\gamma , \alpha}\) on some analytic function spaces is computed in this paper. We study the relation between \(I_f^{\gamma , \alpha}\) and \(J_f^{\gamma , \alpha}\) operators, the \(\beta(\lambda)\) spaces and \(Q_p\) spaces \((0<p<\infty)\).

Blaschke–singular–outer factorization of free non-commutative functions