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The aim of this paper is to prove the theorem which generates many examples of functions belonging to a geometrically defined class of uniformly starlike functions introduced by Goodman in 1991. Only a very few explicit uniformly starlike functions were known until now. Next we obtain inclusion relations between some subclasses of convex functions and the class of uniformly starlike functions. [ABSTRACT FROM AUTHOR]

In the present paper, we found sharp bounds of the second Hankel determinant of logarithmic coefficients of starlike and convex functions of order α . [ABSTRACT FROM AUTHOR]

In this paper, the products of modified Bessel functions of the first kind are considered. Our main aim is to provide various conditions so that these products have certain geometric properties such as starlikeness, convexity, and close to convexity (univalency) with respect to certain functions in the unit disk. Subordinating factor sequences for these products are also obtained. Interesting consequences are provided. The results obtained are new in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sparsity regularization for the diffusion coefficient identification problem. Here, the regularization method is incorporated with the energy functional approach. The advantages of our approach are to deal with convex minimization problems. Therefore, the well-posedness of the problem is obtained without requiring regularity property of the parameter. The convexity of regularized problems also allows use the fast algorithms developed recently. Furthermore, the convergence rates of the method are obtained under a simple source condition. The main results of this paper are the well-posedness and convergence rates of sparsity regularization. We also obtain some new results of the continuity and the differentiability of related operators. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to prove a geometric property for analytic functions p in the open unit disk with $$p(0)=1$$ by using Nunokawa's result, which is a generalized form of well-known Jack's lemma. This property concerns a boundary behavior of the functions p. As the applications of the main result, we obtain a few corollaries where several sufficient conditions for p to be of the real part greater than a given number $$\beta \ (0\le \beta <1)$$ are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Carathéodory function φ Ne (z) = 1 + z - z 3 / 3 maps the open unit disk D onto the interior of the nephroid, a 2-cusped kidney-shaped curve, (u - 1) 2 + v 2 - 4 9 3 - 4 v 2 3 = 0 , and introduce new Ma–Minda-type function classes S Ne ∗ and C Ne associated with it. Apart from studying the characteristic properties of the region bounded by this nephroid, the structural formulas, extremal functions, growth and distortion results, inclusion results, coefficient bounds and Fekete–Szegö problems are discussed for the classes S Ne ∗ and C Ne . Moreover, for β ∈ R and some analytic function p(z) satisfying p (0) = 1 , we prove certain subordination implications of the first-order differential subordination 1 + β z p ′ (z) / p j (z) ≺ φ Ne (z) , j = 0 , 1 , 2 , and obtain sufficient conditions for some geometrically defined function classes available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the sharp bounds for the third Hermitian Toeplitz determinant over classes of functions convex in the direction of the imaginary axis and convex in the direction of the positive real axis are computed. [ABSTRACT FROM AUTHOR]

In this paper, non-differentiable fractional program containing support function of a compact convex set is studied. A class of higher-order cone convex function is introduced which not necessarily lies in any of the existing classes. A non-trivial example of such type of functions is also introduced. Then Schaible dual is formulated for the above-said program for which weak and strong duality theorems are stated and proven using the newly defined function. [ABSTRACT FROM AUTHOR]

In this paper, a class of nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given. [ABSTRACT FROM AUTHOR]

We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and Levin-Stechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality fx+y2≤1(y-x)2∫xy∫xyfs+t2dsdt≤1y-x∫xyf(t)dtsatisfied by every convex function f:R→R and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the weak, strong, and converse duality theorems for a pair of primal-dual problems in set-valued optimization concerning Q-efficient solutions. Then, duality theorems for quasi-relative efficient solutions and Henig efficient solutions are implied with Q being appropriately chosen cones. Finally, their applications to optimality conditions in the Kuhn-Tucker type are obtained. [ABSTRACT FROM AUTHOR]

Let f be analytic in $$D=\{z: |z|< 1\}$$ with $$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$$ . Suppose that $$S^*$$ is the class of starlike functions, and K is the class of close-to-convex functions. The paper instigates a study of finding estimates for Toeplitz determinants whose elements are the coefficients $$a_{n}$$ for f in $$S^*$$ and K. [ABSTRACT FROM AUTHOR]

Following Clunie and Sheil-Small, the class of normalized univalent harmonic mappings in the unit disk is denoted by $${\mathcal {S}}_{{\mathcal {H}}}$$ . The aim of the paper is to study the properties of a subclass of $${\mathcal {S}}_{{\mathcal {H}}}$$ , such that the analytic part is a convex function. We establish estimates of some functionals and bounds of the Bloch's constant for co-analytic part. [ABSTRACT FROM AUTHOR]

It is well known that harmonic convolution of two normalized right half-plane mappings is convex in the direction of the real axis, provided the convolution function is locally univalent and sense-preserving in $$E = \{z: |z|<1\}$$ . Further, it is also known that the condition of local univalence and sense-preserving in E on the convolution function can be dropped when one of the convoluting functions is the standard right half-plane mapping with dilatation $$\displaystyle -z$$ and other is the right half-plane mapping with dilatation $$e^{i\theta } z^n,\,n=1,2,$$ $$\theta \in \mathbb {R}.$$ This result does not hold for $$n=3,4,5,\ldots .$$ In this paper, we generalize this result by taking the dilatation of one of the right half-plane mappings as $$e^{i\theta } z^n$$ $$(\,n\in \mathbb {N},\theta \in \mathbb {R})$$ and that of the other as $$\displaystyle {(a-z)}/{(1-az)},$$ $$a\in (-1,1).$$ We shall prove that our result holds true for all $$n\in \mathbb {N},$$ provided the real constant a is restricted in the interval $$\left[ {(n-2)}/{(n+2)},1\right) $$ . The range of the real constant a is shown to be sharp. [ABSTRACT FROM AUTHOR]

The authors have retracted this article because the article contains major flaws in the proof of the main results. The results of the paper are invalid, since the assumption that the functionals considered are rotationally invariant is not valid. All authors have agreed to this retraction. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of positive solutions for a system of second-order discrete boundary value problem. The main results are obtained via Jensen's inequalities, properties of concave and convex functions, and the Krasnosel'skii-Zabreiko fixed point theorem. Furthermore, concave and convex functions are employed to emphasize the coupling behaviors of nonlinear terms $$f$$ and $$g$$ and we provide two explicit examples to illustrate our main results and the coupling behaviors. [ABSTRACT FROM AUTHOR]

In this paper a(x)-convex functions are defined and Jensen, Hermite-Hadamard, Lah-Ribarić and other inequalities for them are derived. Related analogous of the Lagrange and the Cauchy mean value theorems are also obtained and means of the Cauchy type are generated. [ABSTRACT FROM AUTHOR]

Functions with fixed initial coefficient have been widely studied. A new methodology is proposed in this paper by making appropriate modifications and improvements to the theory of second-order differential subordination. Several interesting examples are given. The results obtained are applied to the classes of convex and starlike functions with fixed second coefficient. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to obtain subordination- and superordination-preserving properties for a class of integral operators defined on the space of normalized analytic functions in the open unit disk. We also consider the sandwich-type theorems for these integral operators. [ABSTRACT FROM AUTHOR]

In this paper, to find a common element of the fixed point set of common fixed points of a countable family of nonexpansive mappings and the solution set of the variational inequality for α-inverse-strongly monotone, we introduce an iterative approximation method in a real Hilbert space. Then the strong convergence theorem is proved under some appropriate conditions imposed on the parameters. This result extended and improved the corresponding results of Yao and Yao [15] and many others. [ABSTRACT FROM AUTHOR]

In this note we describe some results concerning upper and lower bounds for the Jensen functional. We use several known and new results to shed light on the concept of a strongly convex function. [ABSTRACT FROM AUTHOR]

We give an integral version and a refinement of M. Niezgoda's extension of the variant of Jensen's inequality given by A. McD. Mercer. [ABSTRACT FROM AUTHOR]

In this article, we consider the family of functions f analytic in the unit disk | z | < 1 with the normalization f (0) = 0 = f ′ (0) - 1 and satisfying the condition | (z / f (z) ) 2 f ′ (z) - 1 | < λ for some 0 < λ ≤ 1 . We denote this class by U (λ) and we are interested in the relations between U (λ) and other families of functions holomorphic or harmonic in the unit disk. Our first example in this direction is the family of functions convex in one direction. Then, we are concerned with the subordinates to the function 1 / ((1 - z) (1 - λ z)) . We prove that not all functions f(z)/z (f ∈ U (λ)) belong to this family. This disproves an assertion from [14]. Further, we disprove a related coefficient conjecture for U (λ) . We consider the intersection of the class of the above subordinates and U (λ) concerning the boundary behaviour of its functions. At last, with the help of functions from U (λ) , we construct functions harmonic and close-to-convex in the unit disk. [ABSTRACT FROM AUTHOR]

For the exponent ζ > 1 , the diamond alpha Bennett–Leindler type inequalities are established by developing two methods, one of which is based on the convex linear combinations of the related delta and nabla inequalities, while the other one is new and is implemented by using time scale calculus rather than algebra. These inequalities can be considered as the complementary to the classical ones obtained for 0 < ζ < 1. Since both methods provide different diamond alpha Bennett–Leindler type inequalities, we can obtain various diamond alpha unifications of the known delta and nabla Bennett–Leindler type inequalities. Moreover, the second method offers new Bennett–Leindler type inequalities even for the special cases such as delta and nabla ones. Moreover, an application of dynamic Bennett–Leindler type inequalities to the oscillation theory of the second-order half linear dynamic equation is developed and presented for the first time ever. [ABSTRACT FROM AUTHOR]

We extend the well-known open-door lemma by using the theory of differential subordination for functions with fixed initial coefficient and apply it to study some integral operators. Further, using an extension of Nunokawa lemma, we determine some sufficient conditions for the radii and order of starlike functions with fixed second coefficient. Our results improve and generalize some previously known results. [ABSTRACT FROM AUTHOR]

The main goal of this article is to present a new approach, made up of integrals, to refining convex functions inequalities, with applications to several known inequalities like Jensen, Hermite–Hadamard and means inequalities. The first main result will be a Hilbert space operator version, which will be utilized to obtain the aforementioned refinements. In the end, we present graphical interpretations of the new results we have. [ABSTRACT FROM AUTHOR]

The present article concerns the Bohr radius for K-quasiconformal sense-preserving harmonic mappings f = h + g ¯ in the unit disk D for which the analytic part h is subordinated to some analytic function φ , and the purpose is to look into two cases: when φ is convex, or a general univalent function in D . The results state that if h (z) = ∑ n = 0 ∞ a n z n and g (z) = ∑ n = 1 ∞ b n z n , then ∑ n = 1 ∞ (| a n | + | b n |) r n ≤ dist (φ (0) , ∂ φ (D)) for r ≤ r ∗ and give estimates for the largest possible r ∗ depending only on the geometric property of φ (D) and the parameter K. Improved versions of the theorems are given for the case when b 1 = 0 and corollaries are drawn for the case when K → ∞ . [ABSTRACT FROM AUTHOR]

Recent investigations into what geometric properties are preserved under the convolution of two planar harmonic mappings on the open unit disk $${\mathbb {D}}$$ have typically involved half-plane and strip mappings. These results rely on having a convolution that is locally univalent and sense-preserving on $${\mathbb {D}}$$ , and thus, much focus has been on trying to satisfy this condition. We introduce a family of right half-strip harmonic mappings, $$\Psi _c : {\mathbb {D}}\rightarrow {\mathbb {C}}$$ , $$c>0$$ , and consider the convolution $$\Psi _c * f$$ for a harmonic mapping $$f = h +\overline{g}: {\mathbb {D}}\rightarrow {\mathbb {C}}$$ . We prove it is sufficient for $$h \pm g$$ to be starlike for $$\Psi _c *f$$ to be locally univalent and sense-preserving. Moreover, $$\Psi _c * f$$ decomposes into a convex combination of two harmonic mappings, one of which is f itself. This decomposition is key in addressing mapping properties of the convolution, and from it, we produce a family of convex octagonal harmonic mappings as well some other families of convex harmonic mappings. Additionally, motivated by the construction of $$\Psi _c$$ , we introduce a generalized harmonic Bernardi integral operator. We demonstrate convolution preserving properties and a weak subordination relationship for this extended operator. [ABSTRACT FROM AUTHOR]

For real $$\alpha $$ and $$\beta $$ such that $$0 \le \alpha {<} 1 {<} \beta $$ , we denote by $$\varSigma _{s}(\alpha ,\beta )$$ and $$\varSigma _{c}^{0}(\alpha ,\beta )$$ the class of meromorphic univalent functions f such that $$\alpha < Re \left\{ zf'(z)/f(z) \right\} {<} \beta $$ in $$\varDelta $$ and $$\alpha {<} \mathfrak {R}\left\{ f'(z) \right\} {<} \beta $$ in $$\varDelta $$ , where $$\varDelta {=} \left\{ z \in \mathbb {C} : 1<|z|<\infty \right\} $$ , respectively. In the present investigation, estimates on the coefficients of functions of the above classes are investigated. Furthermore, we consider the Fekete-Szegö problem for functions in the above classes. Also, the estimates of the initial coefficients of functions from some subclasses of meromorphic bi-univalent functions are given. [ABSTRACT FROM AUTHOR]

Let $$-1 \le B

Let $$\mathcal {U}$$ denote the class of normalized analytic functions $$f$$ in the open unit disk $$\mathbb {D}$$ satisfying The $$\mathcal {U}$$ -radius is obtained for several classes of functions. These include the class of normalized analytic functions $$f$$ satisfying the inequality $${{\mathrm{Re}}}\, f(z)/g(z)>0$$ or $$\left| f(z)/g(z)-1\right| < 1$$ in $$\mathbb {D},$$ where $$g$$ belongs to a certain class of functions, the class of functions $$f$$ satisfying $$|f'(z)-1|<1$$ in $$\mathbb {D},$$ and functions $$f$$ satisfying $${{\mathrm{Re}}}\, f(z)/z>\alpha , 0 \le \alpha < 1,$$ in $$\mathbb {D}.$$ A recent conjecture by Obradović and Ponnusamy concerning the radius of univalence for a product involving univalent functions is also shown to hold true. [ABSTRACT FROM AUTHOR]

In this work, the generalized Bessel functions with their normalization are considered. Various conditions are obtained so that these Bessel functions have certain geometric properties including close-to-convexity (univalency), starlikeness and convexity in the unit disc. Results obtained for certain classes are new and for the other classes for which similar results exist in the literature, examples are given to support that these results are better than the existing ones. [ABSTRACT FROM AUTHOR]

In this work, we propose a dynamical system (state space) model approach to find a unique minimum of quadratic programming (QP) problems with equality constrained. The unique minimum of the optimization problem is also proved to be asymptotically stable equilibrium point of the state space model. To obtain the optimal solution of QP optimization problem, we seek the limit point of the solution of the state space model by using the transfer function rather than discretization scheme. The numerical results are shown that the applicability and efficiency of the approach by compared with sequential quadratic programming (SQP) method in three examples. [ABSTRACT FROM AUTHOR]

The general classes of multivalent starlike, convex, close-to-convex and quasi-convex functions are introduced. These classes provide a unified treatment to various known subclasses. Inclusion and convolution properties are derived using the methods of convex hull and differential subordination. [ABSTRACT FROM AUTHOR]