Search

Let $K$ be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in $K$, but when $K$ is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) \cite{2}, \cite{13} and \cite{17}. Here, we suppose the field $K$ topologically separable (example $\Bbb C_p$). Then, we give a new definition of strictly analytic functions over a large class of domains called analoid sets. Our theory uses the notion of $T$-sequence which caracterizes analytic sets in the sense of Robba. Thereby we obtain analytic functions satisfying the property of analytic continuation and which, however, will admit expansion in power series (resp. Laurent series) in any disk (resp. in any annulus). Moreover, the algebra of analytic functions will be stable by derivation. The process consists of defining a large class of analytic sets $D$, and a class of admissible sets making a covering of such a $D$, so that we obtain a sheaf on $D$. We finally give an example of differential equation whose solutions are strictly analytic functions in an analoid set. Such an example might not be involved in theories based on affinoid sets.

In this paper, we obtain initial coefficient bounds for functions belong to a subclass of analytic bi-univalent functions related to pseudo-starlike functions by using the Chebyshev polynomials and also we find Fekete-Szego inequalities for this class.

In the present paper, we introduce a measure of the non-univalency of an analytic function, and we use it in order to find the best approximation of analytic function by a locally univalent normalized analytic function.

We introduce a notion of Denjoy sub-system that generalizes that of the Aubry–Mather set. For such systems, we prove a result similar to Denjoy theorem (non-existence of C 2 Denjoy sub-systems), and study their Lyapunov exponents.

It is well known that functions of bounded variation, BV[a, b], and continuous functions, C[a, b], define two classes of functions which are adjoint with respect to the Riemann–Stieltjes integral. This means that: (1) if the integral $$\int _a^b f\,dg$$ exists for every function g in BV[a, b], then f belongs to C[a, b]; (2) if the integral $$\int _a^b f\,dg$$ exists for every function f in C[a, b], then g belongs to BV[a, b]. In this paper, we analyse whether the Kurzweil–Stieltjes integral mimics such a property of the Riemann–Stieltjes integral with respect to some classes of functions. In particular, we investigate the validity of (1) and (2) in the case when the integral is the Kurzweil–Stieltjes integral and C[a, b] is replaced by the class of regulated functions G[a, b] (i.e., functions with discontinuities of the first kind).

We determine the Bohr radius for the class of odd functions $f$ satisfying $|f(z)|\le 1$ for all $|z, 8 pages

In this note, we improve the idea of the Tsallis entropy in a complex domain. This improvement is contingent on the fractional operator in a complex domain (type Alexander). We clarify some new classes of analytic functions, which are planned in view of the geometry function theory. This category of entropy is called fractional entropy; accordingly, we demand them fractional entropic geometry classes. Other geometric properties are established in the sequel. Our exhibition is supported by the Maxwell Lemma and Jack Lemma.

We characterize the functions of some Carleman classes on the unit interval [?1,1] as sums of holomorphic functions in specific neighborhoods of [?1,1]. As an application of our main theorem, we perform an alternative construction of the Dyn?kin?s pseudoanalytic extension for these Carleman classes on [?1,1].

Natalia R. Perelman, Smolensk State University, 4, Przheval’skogo st., 214000, Smolensk, Russia, nataly@mannet.ru The article investigates the three-element Carleman boundary value problem in the class of analytic functions, continuous extension to the contour in the Holder sense, when this problem can not be reduced to a two-element boundary value problems . The unit circle is considered as the contour .To be specific, we study a case of inverse shift. In this case, the solution of the problem is reduced to solving a system of two integral equations of Fredholm second kind; thus significantly used the theory of F. D. Gakhov about Riemann boundary value problem for analytic functions. Based on this result, an algorithm for the solution of the problem is built. Then it is proved that if the boundary condition coefficients are rational functions , and shift a linear-fractional function, then the boundary value problem is solved in an explicit form (in quadrature).Then we consider a simple case of an explicit solution of the problem, when in addition to the above restrictions on the coefficients and shift function is required also analytic continuation of some functions defined on the contour, inside the area. This case is illustrated by a concrete example.

We obtain a uniqueness theorem for a class of analytic functions of exponential type in a halfplane.

In this paper, we introduced two interesting subclasses of the function class σ of analytic and bi-univalent functions in the open unit disk U . Estimates on the first two Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions belonging to these classes are determined. Certain special cases are also indicated.

We consider linear equations for analytic functions in the complex plane with cuts along a half of the boundary of a quadrangle. We propose a regularization method that reduces the equations to an equation with summary-difference kernels. Some applications are given to the moment problem for entire functions of exponential type.

In this paper we introduce and study a new class of analytic and p-valent functions involving higher-order derivatives. For this p-valent function class, we derive several interesting properties including (for example) coefficient inequalities, distortion theorems, extreme points, and the radii of closeto-convexity, starlikeness and convexity. Several applications involving an integral operator are also considered. Finally, we obtain some results for the modified Hadamard product of the functions belonging to the p-valent function class which is introduced here.

This review is dedicated to some recent results on Weyl theory, inverse problems, evolution of the Weyl functions and applications to integrable wave equations in a semistrip and quarter-plane. For overdetermined initial-boundary value problems, we consider some approaches, which help to reduce the number of the initial-boundary conditions. The interconnections between dynamical and spectral Dirac systems, between response and Weyl functions are studied as well., Comment: This paper is a review of recent results, including results from our papers: arXiv:1112.1325, arXiv:1401.3605, arXiv:1403.8111, arXiv:1405.3500 and arXiv:1507.00032

From the point of view of analytic complexity theory, all harmonic functions of two variables split into three classes: functions of complexity zero, one, and two. Only linear functions of one variable have complexity zero. This paper contains a complete description of simple harmonic functions, i.e., of functions of analytic complexity one. These functions constitute a seven-dimensional family expressible as integrals of elliptic functions. All other harmonic functions have complexity two and are, in this sense, of higher complexity. Solutions of the wave equation, the heat equation, and the Hopf equation are also studied.

Let A be the class of analytic functions f defined in the open unit disk E and normalized by f(0)=f'(0)-1=0. For f(z)/z≠0 in E, let Mα,λ:={f∈A:|-αz2(z/f(z))′′ +f′(z)(z/f(z))2-1|≤λ, z∈E}, where λ>0 and α∈R∖[-1/2,0]. In the present paper, we find conditions under which functions in the class M(α,λ) are starlike of order γ, 0≤γ<1.

In this paper, we define the class b Lbα of normalized analytic functions of complex order in the open unit disk, sufficient condition involving coefficient inequalities for f(z) to be in the class b Lbα of analytic functions is obtained. Our main results contain some interesting corollaries as special cases

In this chapter we connect finite Blaschke products to the class of rational functions f such that $$\displaystyle f(\zeta ) \in \widehat {\mathbb {R}}, \quad \zeta \in \mathbb {T}, $$ where we recall that \(\widehat {\mathbb {R}} = \mathbb {R} \cup \{\infty \}\) is the extended real line, regarded as a subset of the Riemann sphere \(\widehat {\mathbb {C}} =\mathbb {C} \cup \{\infty \}\).

The purpose of the present paper is to study certain radii problems for the function , where is a positive real number, is a complex number such that and the function F(z) varies various subclasses of analytic functions with fixed second coefficients. Relevant connections of the results presented herewith various well-known results are briefly indicated.

We prove a Nullstellensatz result for global ideals of Denjoy-Carleman functions in both finitely generated and infinitely generated cases., Comment: 5 pages

In this paper, we introduced a new class of p-valent analytic functions using a linear multiplier Dziok-Srivastava operator D m;q;s f(z)(m2 N0 =f0; 1;:::g

A subclass $\mathcal{A}(n,k)$ of analytic functions $f(z)$ in the unit disc $\mathbb{U}$ is considered. By means of the result due to K. Sakaguchi (J. Math. Soc. Japan {\bf 11}(1959), 72 -- 75) for $f(z) \in \mathcal{A}(1,1)$, some generalization properties of $f(z) \in \mathcal{A}(n,k)$ with several applications are discussed.

For functions $f(z)=z^p+a_{n+1}z^{p+1}+\cdots$ defined on the open unit disk, the condition $RE (f'(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity are investigated and relevant connections with previously known results are indicated.

Our objective in this paper is to consider some basic properties of the familiar Chebyshev polynomials in the theory of analytic functions. We investigate some basic useful characteristics for a class H ( t ) , t ∈ ( 1 / 2 , 1 ] , of functions f , with f ( 0 ) = 0 , f ′ ( 0 ) = 1 , analytic in the open unit disc U = { z : | z | 1 } satisfying the condition that 1 + z f ″ ( z ) f ′ ( z ) ≺ H ( z , t ) = 1 1 − 2 t z + z 2 ( z ∈ U ) , where H ( z , t ) is the generating function of the second kind of Chebyshev polynomials. The Fekete–Szego problem in the class is also solved.

In this paper, we introduce a subclass of analytic functions by using the subordination concept between this function and generalized derivative operator. Some interesting properties of this class are obtained. Article DOI: https://dx.doi.org/10.20319/mijst.2016.s11.318324 This work is licensed under the Creative Commons Attribution-Non-commercial 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.

In this paper, we introduce and investigate a subclass of analytic and bi-univalent functions in the open unit disk U. By using the Faber polynomial expansions, we obtain upper bounds for the coefficients of functions belonging to this analytic and bi-univalent function class. Some interesting recent developments involving other subclasses of analytic and bi-univalent functions are also briefly mentioned.

International audience; A Carleman estimate for a differential operator $P$ is a weighted energy estimate with a weight of exponential form $\exp(\tau \varphi)$ that involves a large parameter, $\tau>0$. The function $\varphi$ and the operator $P$ need to fulfill some sub-ellipticity properties that can be achieved for instance by choosing $\varphi = \exp(\alpha \psi)$, involving a second large parameter, $\alpha>0$, with $\psi$ satisfying some geometrical conditions. The purpose of this article is to give the framework to keep explicit the dependency upon the two large parameters in the resulting Carleman estimates. The analysis is absed on the introduction of a proper Weyl-Hörmander calculus for pseudo-differential operators. Carleman estimates of various strengths are considered: estimates under the (strong) \pcty condition and estimates under the simple characteristic property. In each case the associated geometrical conditions for the function $\psi$ is proven necessary and sufficient. In addition some optimality results with respect to the power of the large parameters are proven.

The object of the present paper is to discuss some interesting properties of analytic functions f(z) associated with the subclasses D( 1; 2; 3; ); G(; ) and Q(; ): Also, radius problems of 1 f(z ) for f(z) in the class D( 1; 2; 3; ); G(; ) and Q(; ) are considered.

Let A denote the family of all analytic functions f in the unit disk D with the normalization f ( 0 ) = 0 = f ' ( 0 ) - 1 . In this note, we mainly consider the radius of univalence of F defined by F ( z ) = z 2 / f ( z ) , where f belongs to some subclasses of A or S , the class of univalent functions from A . The results of the present article sharpen the earlier known results.

In the present paper, the author investigates some subordination and superordination -preserving properties for analytic functions associated with generalized multiplier transformations dened on the space of normalized analytic functions in the open unit disk U. Several Sandwich-type results associated with this transformations is also derived.

In this paper, third-order differential subordination results are obtained for analytic functions associated with an operator defined by the normalized form of the generalized Mittag-Leffler functions. Some particular cases involving Mittag-Leffler and hyperbolic functions are also considered.

In this paper, the Fekete-Szego problem is considered for f ∈ B(λ,α,0) when μ is complexIn this paper, the Fekete-Szego problem is considered for f ∈ B(λ,α,0) when μ is complex

For normalized analytic functions f in the unit disk, the estimate of the integral means L1(r;f ) := r 2 is important in certain problems in fluid dynamics, especially when the functions f (z) are non-vanishing in the punctured unit disk 0 < |z| < 1. We consider the problem of finding the extremal function f which maximizes the integral means L1(r;f ). In addition, for certain class F of analytic functions, we solve the ex- tremal problem for the Yamashita functional

Let \(\mathbb{K}\) be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. Results on branched values obtained in a previous paper are used to prove that algebraic functional equations of the form gq = hfq + w have no solution among transcendental entire functions f, g or among unbounded analytic functions inside an open disk, when w is a polynomial or a bounded analytic function and h is a polynomial or an analytic function whose zeros are of order multiple of q. We also show that an analytic function whose zeros are multiple of an integer q inside a disk is the q-th power of another analytic function, provided q is a prime to the residue characteristic.

It is shown that Denjoy-Carleman quasi-analytic rings of germs of functions in two or more variables fail to satisfy the Weierstrass Preparation Theorem. The result is proven via a non-extension theorem., Comment: 17 pages; accepted at the Advances in Mathematics

R. M. EL-Ashwah Department of Mathematics, Faculty of Science, University of Damietta New Damietta 34517, Egypt. e-mail: r_elashwah@yahoo.com M. K. Aouf Department of Mathematics, Faculty of Science, University of Mansoura Mansoura 35516, Egypt. e-mail: mkaouf127@yahoo.com F. M. Abdulkarem Department of Mathematics, Faculty of Science, University of Damietta New Damietta 34517, Egypt. e-mail: fammari76@gmail.com

We study the radial boundary behavior of functions analytic in a unit disk of the complex plane. 1. The classical Hardy–Littlewood theorem [1] describes the relationship between the smoothness of boundary values of an analytic function on the boundary of a disk of analyticity and the rate of increase in the modulus of its higher-order derivatives. This theorem is an efficient tool for the solution of numerous problems of the theory of functions and the theory of trigonometric series. However, it is often necessary to estimate higher-order derivatives of an analytic function by using solely the information about the modulus of continuity of boundary values of the real part of this function. In the present paper, we consider the following problem: Let a function f be analytic in the disk D := {z ∈ C: |z| < 1} and let the function u := Re f be continuous in D. It is known that, for given n ∈ Z+, the function t �→u(e it ) can be represented in the form

For 0 ≤ α < 1, the sharp radii of starlikeness and convexity of order α for functions of the form f(z) = z + a 2 z 2 + a 3 z 3 + ... whose Taylor coefficients a n satisfy the conditions |a 2| = 2b, 0 ≤ b ≤ 1, and |a n| ≤ n, M or M/n (M > 0) for n ≥ 3 are obtained. Also a class of functions related to Carathéodory functions is considered.

In this paper, we investigate some argument properties for analytic functions with fixed second coefficient and positive real part. And we apply the argument properties to the functions that are analytic and normalized. In particular, the order of strongly starlikeness of strongly convex functions with fixed second coefficients is given.

We obtain some properties related to the coefficient bounds for certain subclass of analytic functions. We also work on the differential subordination for a certain class of functions.