The univalence of α‐project starlike functions

We solve the univalence problem in the class of α‐project starlike function. The α‐project close‐to‐convex function, α‐project Bazilevi? and α‐project Φ‐like functions are also considered here.     

On order of strongly starlikeness in the class of uniformly convex functions

In this paper we consider the order of strongly starlikeness in the classes uniformly convex functions.     

The order of starlikeness of the class of uniformly starlike functions

We investigate classes of k‐uniformly starlike functions which generalize the class UST introduced by Goodman in 1991. Obtained results enable us to solve the problem of finding the largest such that the class of k‐uniformly starlike functions is contained in the class of starlike functions of order α.     

Geometric properties of the functions Γ and 1/Γ

In this paper we determine the radius of convexity and the radius of starlikeness of the functions Γ and The basic tools of our work are the developments of the functions in function series.     

Close‐to‐convexity of normalized Dini functions

In this paper necessary and sufficient conditions are deduced for the close‐to‐convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.     

Hermite‐Hadamard‐type inequalities for interval‐valued coordinated convex functions involving generalized fractional integrals

In this paper, we define interval‐valued left‐sided and right‐sided generalized fractional double integrals. We establish inequalities of Hermite‐Hadamard like for coordinated interval‐valued convex functions by applying our newly defined integrals.     

Hyperbolically convex functions and the generalized Fekete-Szegö functional

We describe extremal functions for the generalized Fekete-Szegö functional over the class of hyperbolically convex functions. We apply the Julia variational formula to reduce the problem to mappings onto hyperbolic polygons having no more than two proper sides. In general, this is the best result possible. We show that both one-sided and two-sided maps can be extremal for different sub-classes.     

单叶解析函数的几个子族之间的关系及应用

在复平面单位圆盘内引入了β型螺形甬数族S<,β>的一个子类S<'β>,<,α>函数族,研究了S<'β>,<,α>族与解析函数族S*,S*(α),K,K(α)及S<,β>之间的关系,利用得到的关系式对S<'β>,<,α>族的第二项系数进行了精确估计,同时得到了K(a)族的第二、三项系数的关系式和S<,β>,族的一个积分表...     

On the stability of a quadratic Jensen type functional equation

In this paper we obtain the general solution of the quadratic Jensen type functional equation

and prove the stability of this equation in the spirit of Hyers, Ulam, Rassias, and G vruta.     

Specht's ratio and complementary inequalities to Golden–Thompson type inequalities on the Hadamard product

As a converse of the arithmetic–geometric mean inequality, W. Specht [Math. Z. 74 (1960) 91–98] estimated the ratio of the arithmetic mean to the geometric one. In this paper, we shall show complementary inequalities to the matricial generalization of Oppenheim's inequality and the Golden–Thompson type inequalities on the Hadamard product by T. Ando [Linear Algebra Appl. 26 (1979) 203; Linear Algebra Appl. 241–243 (1996) 105], in which Specht's ratio plays an important role. As an application, we shall obtain a complementary inequality to the Hadamard determinant inequality.     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.     

Plancherel–Polya-type inequalities for entire functions of exponential type in

The goal of the paper is to prove generalizations of the classical Plancherel–Polya inequalities in which point-wise sampling of functions (δ-distributions) is replaced by more general compactly supported distributions on . As an application it is shown that a function , 1p∞, which is an entire function of exponential type is uniquely determined by a set of numbers {Ψ_j(f)}, , where {Ψ_j}, , is a countable sequence of compactly supported distributions. In the case p=2 a reconstruction method of a Paley–Wiener function f from a sequence of samples {Ψ_j(f)}, , is given. This method is a generalization of the classical result of Duffin–Schaeffer about exponential frames on intervals.     

Non‐commutative ‐divergence functional

We introduce the non‐commutative f‐divergence functional for an operator convex function f, where and are continuous fields of Hilbert space operators and study its properties. We establish some relations between the perspective of an operator convex function f and the non‐commutative f‐divergence functional. In particular, an operator extension of Csiszár's result regarding f‐divergence functional is presented. As some applications, we establish a refinement of the Choi–Davis–Jensen operator inequality, obtain some unitarily invariant norm inequalities and give some results related to the Kullback–Leibler distance.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

Limit cycles in a Gause‐type predator–prey model with sigmoid functional response and weak Allee effect on prey

The goal of this work is to examine the global behavior of a Gause‐type predator–prey model in which two aspects have been taken into account: (i) the functional response is Holling type III; and (ii) the prey growth is affected by a weak Allee effect. Here, it is proved that the origin of the system is a saddle point and the existence of two limit cycles surround a stable positive equilibrium point: the innermost unstable and the outermost stable, just like with the strong Allee effect. Then, for determined parameter constraints, the trajectories can have different ω ? limit sets. The coexistence of a stable limit cycle and a stable positive equilibrium point is an important fact for ecologists to be aware of the kind of bistability shown here. So, these models are undoubtedly rather sensitive to disturbances and require careful management in applied contexts of conservation and fisheries. Copyright © 2012 John Wiley & Sons, Ltd.     

Boundedness of solutions to the critical fully parabolic quasilinear one‐dimensional Keller–Segel system

In this paper we consider a one‐dimensional fully parabolic quasilinear Keller–Segel system with critical nonlinear diffusion. We show uniform‐in‐time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well‐known Lyapunov functional and a recently introduced new Lyapunov‐like functional in 3 .     

Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

We prove $L^p$ estimates, $1\le p\le \infty$, for solutions to the tangential Cauchy–Riemann equations ${\overline{?}}_{b}u=?$ on a class of infinite type domains $\mathrm{\Omega }?{\mathbb{C}}^2$. The domains under consideration are a class of convex ellipsoids, and we show that if $?$ is a ${\overline{?}}_b$‐closed (0,1)‐form with coefficients in $L^p$, then there exists an explicit solution u satisfying ${\parallel u\parallel }_{L^p(b\mathrm{\Omega })}\le C{\parallel ?\parallel }_{L^p(b\mathrm{\Omega })}$. Moreover, when $p=\infty$, we show that there is a gain in regularity to an f‐Hölder space. We also present two applications. The first is a solution to the $\overline{?}$‐equation, that is, given a smooth (0,1)‐form ? on $b\mathrm{\Omega }$ with an L¹‐boundary value, we can solve the Cauchy–Riemann equation $\overline{?}u=?$ so that ${\parallel u\parallel }_{L^1(b\mathrm{\Omega })}\le C{\parallel ?\parallel }_{L^1(b\mathrm{\Omega })}$ where C is independent of $u$ and ?. The second application is a discussion of the zero sets of holomorphic functions with zero sets of functions in the Nevanlinna class within our class of domains.     

Metrics of Kaluza–Klein type on the anti‐de Sitter space

We introduce and study a new family of pseudo‐Riemannian metrics on the anti‐de Sitter three‐space . These metrics will be called “of Kaluza‐Klein type” , as they are induced in a natural way by the corresponding metrics defined on the tangent sphere bundle . For any choice of three real parameters , the pseudo‐Riemannian manifold is homogeneous. Moreover, we shall introduce and study some natural almost contact and paracontact structures , compatible with , such that is a homogeneous almost contact (respectively, paracontact) metric structure. These structures will be then used to show the existence of a three‐parameter family of homogeneous metric mixed 3‐structures on the anti‐de Sitter three‐space.     

Hartman–Wintner‐type inequalities for a class of nonlocal fractional boundary value problems

In this paper, we establish new Hartman–Wintner‐type inequalities for a class of nonlocal fractional boundary value problems. As an application, we obtain a lower bound for the eigenvalues of corresponding equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Lyapunov–Razumikhin method for impulsive functional differential equations and applications to the population dynamics

The present paper deals with the investigation of the stability of the zero solution of impulsive functional differential equations. By means of piecewise continuous functions coupled with the Razumikhin technique sufficient conditions for stability, uniform stability and asymptotic stability of the zero solution of such equations are found.