Abstract: We establish the mapping properties of the fractional integral operators with homogeneous kernels on generalized Lorentz-Morrey spaces. Mathematics subject classification (2010): Fractional integral operators, Morrey spaces, Lorentz spaces.

Abstract: In the article, we provide some new post quantum refinements of the Hermite-Hadamard like inequalities involving the class of h -preinvex functions by establishing a new auxiliary result involving the post quantum differentiable function. By discussing some special cases, it is shown that our obtained results are the further generalizations of many previous known results. Mathematics subject classification (2020): 26A51, 26D15, 05A30.

Abstract: In this paper by defining a extended Hardy operator, a new kernel function including both the homogeneous and the non-homogeneous cases is constructed. Dealing with these cases in a unified way, a Hilbert-type inequality involving the newly constructed kernel is established, and the constant factor is proved to be the best possible. The equivalent Hardy-type inequality is also considered in parallel. Furthermore, by specifying the kernel function, some special and meaningful Hilbert-type inequalities with the constant factors related to the higher derivative of trigonometric functions and special functions are presented at the end of the paper, and these newly obtained inequalities are proved to be the extensions of some classical Hilbert-type inequalities.

Abstract: Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called g -composite convexity as a generalization of log convexity. Then we prove that g -composite convex functions have better estimates in certain known inequalities like the Hermite-Hadamard inequality, super additivity of convex functions, the Majorization inequality and some means inequalities. Strongly related to this, we define the index of convexity as a measure of “how much the function is convex”. Applications including Hilbert space operators, matrices and entropies will be presented in the end. Mathematics subject classification (2020): Primary 26A51; Secondary 47A30, 39B62, 26D07, 47B15, 15A60.

Abstract: Using the properties of geometric mean, we shall show for any $0\le \alpha ,\beta \le 1$, \[f\left( A{{ abla }_{\alpha }}B \right)\le f\left( \left( A{{ abla }_{\alpha }}B \right){{ abla }_{\beta }}A \right){{\sharp}_{\alpha }}f\left( \left( A{{ abla }_{\alpha }}B \right){{ abla }_{\beta }}B \right)\le f\left( A \right){{\sharp}_{\alpha }}f\left( B \right)\] whenever $f$ is a non-negative operator log-convex function, $A,B\in \mathcal{B}\left( \mathcal{H} \right)$ are positive operators, and $0\le \alpha ,\beta \le 1$. As an application of this operator mean inequality, we present several refinements of the Aujla subadditive inequality for operator monotone decreasing functions. Also, in a similar way, we consider some inequalities of Ando's type. Among other things, it is shown that if $\Phi $ is a positive linear map, then \[\Phi \left( A{{\sharp}_{\alpha }}B \right)\le \Phi \left( \left( A{{\sharp}_{\alpha }}B \right){{\sharp}_{\beta }}A \right){{\sharp}_{\alpha }}\Phi \left( \left( A{{\sharp}_{\alpha }}B \right){{\sharp}_{\beta }}B \right)\le \Phi \left( A \right){{\sharp}_{\alpha }}\Phi \left( B \right).\]

Abstract: In this paper, we introduce and solve the following additive-additive (s,t) -functional inequality ‖g(x+ y)−g(x)−g(y)‖+‖h(x+ y)+h(x− y)−2h(x)‖ (1) ∥∥∥s ( 2g ( x+ y 2 ) −g(x)−g(y) ∥∥∥+ ∥∥∥t ( 2h ( x+ y 2 ) +2h ( x− y 2 ) −2h(x) ∥∥∥ , where s and t are fixed nonzero complex numbers with |s| < 1 and |t| < 1 . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of derivation-homomorphisms in complex Banach algebras, associated to the additive-additive (s,t) -functional inequality (1) and the following functional inequality ‖g(xy)−g(x)y− xg(y)‖+‖h(xy)−h(x)h(y)‖ φ(x,y). (2) Mathematics subject classification (2010): Primary 47B47, 17B40, 39B72, 39B62, 39B52.

Abstract: In this article, we present a new treatment of the arithmetic-geometric mean inequality and its siblings, the Heinz and the Young inequalities. New refinements via calculus computations and convex analysis are presented and a new Heinz-type inequality is presented for any symmetric operator mean.

Abstract: The main focus of the present paper is to establish definite integral formulae for ratios of the Fox–Wright functions. As consequences of the master formula, some novel integral formulae are derived for ratios of other special functions which are associated to Fox–Wright Ψ function, like generalized hypergeometric function, modified Bessel function of the first kind and Mittag–Leffler type functions of two and three parameters. Moreover, closed integral form expressions are obtained for a family of Mathieu-type series and for the associated alternating versions whose terms contain the incomplete Fox-Wright function. As applications, functional bounding inequalities are established for the aforementioned series.

Abstract: Let A,B,X and Y be n×n complex matrices such that A and B are positive semidefinite, then ‖AX +YB‖ 1 4 (‖W1‖+‖W2‖+W4) , where W1 = A+A1/2 |X∗|2 A1/2, W2 = B+B |Y | B, W3 = AXB +AYB and W4 = √ (‖W1‖−‖W2‖)2 +4‖W3‖. Multiple results are given in this paper.

Abstract: In this paper we introduce and investigate a new generalized class of bi-bazilevič functions defined by using (s,t) -derivative operator and quasi-subordination in the open unit disk D . We obtain two kinds of coefficient estimate by using Faber polynomial expansion and get Fekete–Szegö inequality for the new class and some of its subclasses. Mathematics subject classification (2020): 30C45.

Abstract: In this paper, we give a new generalization of Darbo’s fixed point theorem of integral type. An application for the solvability of nonlinear fractional integral equation is given to illustrate our result.

Abstract: In this paper, we introduce a nonlinear inequality based on four self-mappings. We give necessary conditions which ensure the existence of a common fixed point of four selfmappings satisfying said inequality defined in Ś -metric spaces. A common fixed point problem is discussed. We set up an example to elucidate our main result. Moreover, the existence of a common solution to a system of four integral equations is shown by application of main result. Mathematics subject classification (2020): 47H10, 39B62, 26E05.

Abstract: Using the fixed point method, we prove the Hyers-Ulam stability of homomorphisms in complex Banach algebras and complex Banach Lie algebras and also of derivations on complex Banach algebras and complex Banach Lie algebras for the general additive functional inequality ‖ f (αx− βy)−α f (x)− β f (−y)‖ ‖r( f (αx+ βy)−α f (x)−β f (y)‖ , where r is a fixed nonzero complex number with |r| < 1 and α ,β = 0 . Mathematics subject classification (2010): Primary 39B62, 39B52, 47H10, 46B25.

Abstract: In this work, by the weighted arithmetic-geometric mean inequality, we show if a,b > 0 and 0 ν 1. Then for all positive integer m, we have ( aν b1−ν )m + r 0 ( (a+b) −2m(ab) 2 ) +rm [( (ab) m 4 −b 2 )2 χ(0, 2 ](ν)+ ( (ab) m 4 −a 2 )2 χ( 2 ,1](ν) ] ( νa+(1−ν)b )m , where r0 = min{ν ,1−ν}, rm = min{2mrm 0 ,(1− r0)− rm 0 } and χI(ν) the characteristic function. This inequality provides a generalization of an important refinement of the Young inequality obtained by J. Zhao and J. Wu. As applications we give some new generalized refinements of Young type inequalities for the determinants, p -norms and traces, of positive τ -measurable operators. Mathematics subject classification (2020): 26D07, 26D15, 46L52, 47A63.

Abstract: We propose two modifications for Gauss-Weierstrass operators and moment-type operators which fix eax and e2ax with a > 0. First, we present moment identities for new operators. Then, we discuss weighted approximation and prove Voronovskaya-type theorems for them in exponentially weighted spaces. Using modulus of continuity in exponentially weighted spaces, we obtain some global smoothness preservation properties. We give a comparison result for Gauss-Weierstrass operators. Finally, we provide some graphical illustrations that show that modified operators perform better than classical ones. Mathematics subject classification (2020): Primary 41A35, 41A25; Secondary 47G10.

Abstract: In this article, we study a complete convergence theorem for weighted sums in sublinear expectations space. We establish a complete convergence theorem for weighted sums under the optimal moment conditions in sub-linear expectations space. Our result extends and improves the corresponding result of Cai (Metrika, 68:323-331, 2008) in some extent.

Abstract: In this paper, we generalize Opial inequality to higher dimensions on time scales. We prove the Opial Delta-nabla inequality of n variables, and then give two diamond-alpha dynamic inequalities of Opial type of n variables. As well, we introduce some special cases. Mathematics subject classification (2020): 26E70, 26E25.

Abstract: Let N > 1 be a real number and ε > 0 be given. In this paper, we will prove that, for a measurable subset S of [0,N] with positive density ε , there must be patterns of the form (x,x+ t,x+ γ(t)) such that x,x+ t,x+ γ(t) ∈ S, where γ is convex and has some curvature constraints, t > δ (ε ,γ)γ−1(N) and δ (ε ,γ) is a positive constant depending only on ε and γ , γ−1 is the inverse function of γ . Our result extends Bourgain’s result [2] to the general curve γ . We use Bourgain’s energy pigeonholing argument and Li’s σ -uniformity argument.