On certain classes of analytic functions defined by Noor integral operator

Integral operator, introduced by Noor, is defined by using convolution. Let f_n(z)=z/(1−z)ⁿ⁺¹, n∈N₀, and let f be analytic in the unit disc E. Then I_nf=f⁽⁻¹⁾_n★f, where f_n★f⁽⁻¹⁾_n=z/(1−z). Using this operator, certain classes of analytic functions, related with the classes of functions with bounded boundary rotation and bounded boundary radius rotation, are defined and studied in detail. Some basic properties, rate of growth of coefficients, and a radius problem are investigated. It is shown that these classes are closed under convolution with convex functions. Most of the results are best possible in some sense.     

Eigenvalue inequalities for convex and log-convex functions

We give a matrix version of the scalar inequality f(a + b) ? f(a) + f(b) for positive concave functions f on [0, ∞). We show that Choi’s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences. Some inequalities for log-convex functions are presented and a new arithmetic-geometric mean inequality for positive matrices is given. We also point out a simple proof of the Bhatia-Kittaneh arithmetic-geometric mean inequality.     

On monotone functions of tree structures

Let T be a rooted tree structure with n nodes a₁,…,a_n. A function f: {a₁,…,a_n} into {1 < ? < k} is called monotone if whenever a_i is a son of a_j, then f(a_i) ≥ f(a_j). The average number of monotone bijections is determined for several classes of tree structures. If k is fixed, for the average number of monotone functions asymptotic equivalents of the form c · ?^?nn^{?$\text{3}\text{2}$} (n → ∞) are obtained for several classes of tree structures.     

Uniqueness of entire functions

Let f be a nonconstant entire function and let a be a meromorphic function satisfying T(r,a)=S(r,f) and a?a′. If f(z)=a(z)⇔f′(z)=a(z) and f(z)=a(z)⇒f″(z)=a(z), then f≡f′, and a?a′ is necessary. This extended a result due to Jank, Mues and Volkmann.     

On functions convex in the direction of the real axis with a fixed second coefficient

In the paper we consider the class Γ of analytic and univalent functions f in the unit disk Δ, normalized by f(0) = f′(0) − 1 = 0, having real coefficients and such that f(Δ) is convex in the direction of the real axis. We are especially interested in some subclasses of Γ. The most important of them is Γ(c) consisting of those functions which have the second coefficients of the Taylor expansion fixed and equal to c. We obtain the Koebe set for this class as well as for the classes Γ⁺(c) and Γ⁻(c) of functions which are in some sense convex in the direction of positive and negative axes respectively.     

Value distribution and shared sets of differences of meromorphic functions

We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n?2, then fn(z)f(z+c) assumes every non-zero value a∈C infinitely often. This research also shows that there exist two sets S₁ with 9 (resp. 5) elements and S₂ with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, Ef(z)(Sj)=Ef(z+c)(Sj)(j=1,2) imply f(z)≡f(z+c). This gives an answer to a question of Gross concerning a finite order meromorphic function f and its shift.     

Log-convexity and log-concavity of hypergeometric-like functions

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a?∑fkk(a)xk, a?∑fkΓ(a+k)xk and a?∑fkxk/k(a). The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.     

The support points of several classes of analytic functions with fixed coefficients

Let B denote the set of functions ?(z) that are analytic in the unit disk D and satisfy |?(z)|?1(|z|<1). Let P denote the set of functions p(z) that are analytic in D and satisfy p(0)=1 and Rep(z)>0(|z|<1). Let T denote the set of functions f(z) that are analytic in D, normalized by f(0)=0 and f^′(0)=1 and satisfy that f(z) is real if and only if z is real (|z|<1). In this article we investigate the support points of the subclasses of B, P and T of functions with fixed coefficients.     

Periodic decomposition of measurable integer valued functions

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a₁,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a₁,…,ak)-periodic decomposition if and only if Δa₁?Δakf=0, where Δaf(x)=f(x+a)−f(x).     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.     

Continuity properties in non-commutative convolution algebras, with applications in pseudo-differential calculus

We study continuity properties for a family {s_p}_p?1 of increasing Banach algebras under the twisted convolution, which also satisfies that a∈s_p, if and only if the Weyl operator a^w(x,D) is a Schatten-von Neumann operator of order p on L². We discuss inclusion relations between the s_p-spaces, Besov spaces and Sobolev spaces. We prove also a Young type result on s_p for dilated convolution. As an application we prove that f(a)∈s₁, when a∈s₁ and f is an entire odd function. We finally apply the results on Toeplitz operators and prove that we may extend the definition for such operators.     

Meromorphic functions sharing two small functions with its derivative

In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM^∗, and share b(z)IM^∗ with g(z)=a₁(z)f(z)+a₂(z)f^′(z). And a₁(z), a₂(z) and b(z) (a₂(z),b(z)?0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.     

Generalization of Berg-Dimovski convolution in spaces of analytic functions

In the space H(G) of functions analytic in a ρ-convex region G equipped with the topology of compact convergence, we construct a convolution for the operator J _π+L where J _ρ is the generalized Gel’fond-Leont’ev integration operator and L is a linear continuous functional on H(G). This convolution is a generalization of the well-known Berg-Dimovski convolution. We describe the commutant of the operator J _π+L in ?(G) and obtain the representation of the coefficient multipliers of expansions of analytic functions in the system of Mittag-Leffler functions.     

Entire functions sharing one small function CM with their shifts and difference operators

In this article, we mainly devote to proving uniqueness results for entire functions sharing one small function CM with their shift and difference operator simultaneously. Let f(z) be a nonconstant entire function of finite order, c be a nonzero finite complex constant,and n be a positive integer. If f(z), f(z + c), and ?_c~n f(z) share 0 CM, then f(z + c) ≡ Af(z),where A(= 0) is a complex constant. Moreover, let a(z), b(z)( ≡ 0) ∈ S(f) be periodic entire functions with period c and if f(z)-a(z), f(z + c)-a(z), ?_c~n f(z)-b(z) share 0 CM, then f(z + c) ≡ f(z).     

Keller-Osserman conditions for diffusion-type operators on Riemannian manifolds

In this paper we obtain essentially sharp generalized Keller-Osserman conditions for wide classes of differential inequalities of the form Lu?b(x)f(u)?(|∇u|) and Lu?b(x)f(u)?(|∇u|)−g(u)h(|∇u|) on weighted Riemannian manifolds, where L is a non-linear diffusion-type operator. Prototypical examples of these operators are the p-Laplacian and the mean curvature operator. The geometry of the underlying manifold is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions b and ?. A weak maximum principle which extends and improves previous results valid for the φ-Laplacian is also obtained. Geometric comparison results, valid even in the case of integral bounds for the modified Bakry-Emery Ricci tensor, are presented.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) ? 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator \({f \mapsto f_c}\) maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) − 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

The structure of rigid functions

A function f:R→R is called vertically rigid if graph(cf) is isometric to graph(f) for all c≠0. We prove Jankovi?'s conjecture by showing that a continuous function is vertically rigid if and only if it is of the form a+bx or a+bekx (a,b,k∈R). We answer the question of Cain, Clark and Rose by showing that there exists a Borel measurable vertically rigid function which is not of the above form. We discuss the Lebesgue and Baire measurable case, consider functions bounded on some interval and functions with at least one point of continuity. We also introduce horizontally rigid functions, and show that a certain structure theorem can be proved without assuming any regularity.     

Orthogonally additive polynomials on spaces of continuous functions

We show that, for every orthogonally additive homogeneous polynomial P on a space of continuous functions C(K) with values in a Banach space Y, there exists a linear operator S:C(K)→Y such that P(f)=S(fn). This is the C(K) version of a related result of Sundaresam for polynomials on Lp spaces.     

Composite entire functions with no unbounded Fatou components

Let L be the set of all entire functions f such that for given ?>0,

logL(r,f)>(1−?)logM(r,f)