Showing papers in "Mathematical Inequalities & Applications in 2020"
TL;DR: In this article, several monotonicity properties and bounds for the complete elliptic integral of the first kind were presented, as well as sharp bounds for arithmetic-geometric mean.
Abstract: In the article, we present several monotonicity properties and bounds for the complete elliptic integral of the first kind. As applications, we find sharp bounds for the arithmeticgeometric mean. Mathematics subject classification (2010): 33E05, 26E60.
63 citations
TL;DR: In this article, the authors presented several new bounds for the complete elliptic integrals K (r)= ∫ π/2 0 (1−r2 sin2 θ )−1/2dθ and E (r), and found an asymptotic expansion for K(r) as r → 1, which are the refinements and improvements of previously well-known results.
Abstract: In the article, we present several new bounds for the the complete elliptic integrals K (r)= ∫ π/2 0 (1−r2 sin2 θ )−1/2dθ and E (r)= ∫ π/2 0 (1−r2 sin2 θ )1/2dθ , and find an asymptotic expansion for K (r) as r → 1 , which are the refinements and improvements of the previously well-known results. Mathematics subject classification (2010): 33E05, 26E60.
31 citations
TL;DR: In this article, the authors prove that the inequality of the inequality is not the same as that of the distribution of the inequalities of the distributions of the nodes in the graph. But
Abstract: In the article, we prove that the inequality
31 citations
26 citations
TL;DR: In this article, the authors established generalizations of Sobolev's theorem for double phase functionals, where Φ(x,t) = t p + {b(x)t(log(e+ t))τ}, where 1 < p q < ∞, τ > 0 and b is a nonnegative bounded function satisfying |b (x)− b(y)| C|x− y|θ (log(m+n)−1))−τ for 0 θ < 1.
Abstract: Our aim in this paper is to establish generalizations of Sobolev’s theorem for double phase functionals Φ(x,t) = t p + {b(x)t(log(e+ t))τ} , where 1 < p q < ∞ , τ > 0 and b is a nonnegative bounded function satisfying |b(x)− b(y)| C|x− y|θ (log(e+ |x− y|−1))−τ for 0 θ < 1 . Mathematics subject classification (2010): 46E30, 42B25, 46E35.
12 citations
11 citations
TL;DR: In this article, the concept of harmonical $h$-convexity for interval-valued functions is introduced and investigated, and some new Hermite-Hadamard type inequalities for the interval Riemann integral are proved.
Abstract: We introduce and investigate the concept of harmonical $h$-convexity for interval-valued functions. Under this new concept, we prove some new Hermite-Hadamard type inequalities for the interval Riemann integral.
9 citations
TL;DR: In this article, the authors presented an explicit formula and an identity for higher order derivatives of exponential polynomials with the help of the Faà di Bruno formula, properties of the Bell polynoms of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds.
Abstract: In the paper, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, the author presents an explicit formula and an identity for higher order derivatives of generating functions of exponential polynomials; consequently, the author recovers an explicit formula and finds an identity for exponential polynomials in terms of the Stirling numbers of the fist and second kinds; furthermore and importantly, with the assistance of the complete monotonicity of generating functions of exponential polynomials and other known conclusions, the author constructs some determinantal inequalities and product inequalities and deduces the logarithmic convexity and logarithmic concavity of two sequences related to exponential polynomials; finally, the author gives an application of exponential polynomials by confirming that exponential polynomials satisfy conditions for sequences required in white noise distribution theory. Mathematics subject classification (2010): 11B83, 11A25, 11B73, 11C08, 11C20, 15A15, 26A24, 26A48, 26C05, 26D05, 33B10, 34A05, 60H40.
9 citations
TL;DR: In this paper, an integral, asymptotic expansion and Maclaurin series representation for the generalized Gaussian ratio is presented, and several simple approximations for its inverse function are presented, which may be essential to the estimations for the shape parameter.
Abstract: We are devoted to an integral, asymptotic expansion and Maclaurin series representation for the generalized Gaussian ratio, and find their various related properties such as complete monotonicity and some useful inequalities. As applications, several simple approximations for its inverse function are presented, which may be essential to the estimations for the shape parameter of the generalized Gaussian distribution. Mathematics subject classification (2010): 26A48, 26D05, 33B10, 33B15, 62F99.
8 citations
TL;DR: In this paper, the authors present simple proofs of Choi's results and give a short alternative proof for Fiedler and Markham's inequality. And they also obtain additional matrix inequalities related to partial determinants.
Abstract: In this paper, we first present simple proofs of Choi's results [4], then we give a short alternative proof for Fiedler and Markham's inequality [6]. We also obtain additional matrix inequalities related to partial determinants.
7 citations
TL;DR: In this article, the best possible constant factor related to some parameters, some particular cases and the operator expressions are considered in the context of subject classification, and a more accurate extended Mulholland's inequality and its equivalent form are given.
Abstract: By the use of the weight functions, the idea of introduced parameters and HermiteHadamard’s inequality, a more accurate extended Mulholland’s inequality and its equivalent form are given. A few equivalent statements of the best possible constant factor related to some parameters, some particular cases and the operator expressions are considered. Mathematics subject classification (2010): 26D15.
TL;DR: In this paper, the authors presented new improvements of certain Cauchy-Schwarz type inequalities for Hilbert space operators and provided refinements of some numerical radius inequalities for operators.
Abstract: We present new improvements of certain Cauchy–Schwarz type inequalities. As applications of the results obtained, we provide refinements of some numerical radius inequalities for Hilbert space operators. It is shown, among other inequalities, that if A ∈ B(H ) , then ω (A) 1 6 ∥ ∥ ∥|A| + |A∗|2 ∥ ∥ ∥+ 1 3 ω (A)‖|A|+ |A∗|‖ . Mathematics subject classification (2010): 47A12, 47A30, 15A60.
TL;DR: In this article, the authors characterize the boundedness and compactness of the operators Tψ1,ψ2,φ from weighted Bergman spaces to weighted-type and little weighted type spaces of analytic functions.
Abstract: Let H (D) be the space of analytic functions on the unit disc D . Let φ be an analytic self-map of D and ψ1,ψ2 ∈H (D) . Let Cφ , Mψ and D denote the composition, multiplication and differentiation operators, respectively. In order to treat the products of these operators in a unified manner, Stević et al. introduced the following operator Tψ1 ,ψ2 ,φ f = ψ1 · f ◦φ +ψ2 · f ′ ◦φ , f ∈ H (D). We characterize the boundedness and compactness of the operators Tψ1 ,ψ2 ,φ from weighted Bergman spaces to weighted-type and little weighted-type spaces of analytic functions. Also, we give examples of bounded, unbounded, compact and non compact operators Tψ1 ,ψ2 ,φ . Mathematics subject classification (2010): 47B33, 47B38.
TL;DR: The authors provided sharp bounds for the q -gamma function from below and above for all q > 0 by means of investigating the monotonicity property to analytical functions involving logarithm q-gamma functions.
Abstract: This paper is devoted to provide sharp bounds for the q -gamma function from below and above for all q > 0 by means of investigating the monotonicity property to analytical functions involving logarithm q -gamma function. It turns out that these results refine and improve lower and upper bounds for the q -gamma function which have been given by Salem [13]. Mathematics subject classification (2010): 33D05, 26D07, 26A48.
TL;DR: In this article, the largest claim amounts of two sets of interdependent portfolios were compared in the sense of usual stochastic order, when the variables in one set have the parameters $\lambda_1,\ldots,\lambda_n$ and $p_ 1, \ldots,p_ n$ and when the variable in the other set has the parameters
Abstract: Let $ X_{\lambda_1},\ldots,X_{\lambda_n}$ be dependent non-negative random variables and $Y_i=I_{p_i} X_{\lambda_i}$, $i=1,\ldots,n$, where $I_{p_1},\ldots,I_{p_n}$ are independent Bernoulli random variables independent of $X_{\lambda_i}$'s, with ${\rm E}[I_{p_i}]=p_i$, $i=1,\ldots,n$. In actuarial sciences, $Y_i$ corresponds to the claim amount in a portfolio of risks. In this paper, we compare the largest claim amounts of two sets of interdependent portfolios, in the sense of usual stochastic order, when the variables in one set have the parameters $\lambda_1,\ldots,\lambda_n$ and $p_1,\ldots,p_n$ and the variables in the other set have the parameters $\lambda^{*}_1,\ldots,\lambda^{*}_n$ and $p^*_1,\ldots,p^*_n$. For illustration, we apply the results to some important models in actuary.
TL;DR: This paper proves a considerable generalization of the two refinements of the arithmetic-geometric mean inequality due to Furuichi, Manasrah and Kittaneh, which correspond to the cases m = 1 and n = 2 , respectively.
Abstract: In this paper, we prove that for i = 1,2, . . . ,n , ai 0 and αi > 0 satisfy ∑i=1 αi = 1 , then for m = 1,2,3, . . . , we have ( n ∏ i=1 ai i )m + rm 0 ( n ∑ i=1 ai −n n √ n ∏ i=1 ai ) ( n ∑ i=1 αiai )m where r0 = min{αi : i = 1, . . . ,n} . This is a considerable generalization of the two refinements of the arithmetic-geometric mean inequality due to Furuichi [2], Manasrah and Kittaneh [7], which correspond to the cases m = 1 and n = 2 , respectively. As application we give some generalized inequalities of determinants for positive definite matrices. Mathematics subject classification (2010): 26D07, 26D15, 15A45.
TL;DR: In this paper, the authors set up several identities that imply some versions of the Hardy type inequalities and provided the virtual extremizers for many Hardy types inequalities, giving a straightforward understanding of several Hardy types and the nonexistence of nontrivial optimizers.
Abstract: We set up several identities that imply some versions of the Hardy type inequalities. These equalities give a straightforward understanding of several Hardy type inequalities as well as the nonexistence of nontrivial optimizers. These identities also provide the “virtual” extremizers for many Hardy type inequalities. Mathematics subject classification (2010): 26D10, 46E35, 35A23.
TL;DR: In this paper, the authors considered the space of polynomials of degree at most three in the real line endowed with the sup norm over the unit interval and provided, explicitly, all the extreme points of the unit ball of this space.
Abstract: In this paper we consider the space of polynomials of degree at most three in the real line endowed with the sup norm over the unit interval. We provide, explicitly, all the extreme points of the unit ball of this space. Using the previous geometrical description, we obtain the Bernstein function for the first and second derivative of the polynomials of degree at most 3. Mathematics subject classification (2010): 41A17, 26D05.
TL;DR: In this article, the authors characterize Hardy inequality in weighted Lebesgue sequence spaces involving discrete bilinear Hardy operator (n ∑ i=−∞ ai )( n ∑ ii=− ∞ bi ) and then apply this information to characterize the inequality with q -bilinear Hardiness operator Hq( f,g)(x) := (∫ ∞ 0 χ(0,x), f (t)dqt )(∫∞ 0,x ],t)g(t)Dqt ) for all possible indices of
Abstract: We characterize Hardy inequality in weighted Lebesgue sequence spaces involving discrete bilinear Hardy operator ( n ∑ i=−∞ ai )( n ∑ i=−∞ bi ) and then we apply this information to characterize the inequality with q -bilinear Hardy operator Hq( f ,g)(x) := (∫ ∞ 0 χ(0,x](t) f (t)dqt )(∫ ∞ 0 χ(0,x](t)g(t)dqt ) for all possible indices of summation. Mathematics subject classification (2010): 26D10, 46E35.
TL;DR: Lower and upper bounds on the incomplete gamma function were given in this paper for all real $a$ and all real$x>0 for all values of $x, away from $0$ and $infty.
Abstract: Lower and upper bounds $B_a(x)$ on the incomplete gamma function $\Gamma(a,x)$ are given for all real $a$ and all real $x>0$. These bounds $B_a(x)$ are exact in the sense that $B_a(x)\underset{x\downarrow0}\sim\Gamma(a,x)$ and $B_a(x)\underset{x\to\infty}\sim\Gamma(a,x)$. Moreover, the relative errors of these bounds are rather small for other values of $x$, away from $0$ and $\infty$.
TL;DR: In this article, it was shown that the local block space with variable exponents is a pre-dual of the local Morrey space with variables, and the extrapolation theory for the local sharp maximal functions, the geometric maximal functions and the rough maximal function on the local MORrey spaces with variables was established.
Abstract: We study the local Morrey spaces with variable exponents. We show that the local block space with variable exponents are pre-duals of the local Morrey spaces with variable exponents. Using this duality, we establish the extrapolation theory for the local Morrey spaces with variable exponents. The extrapolation theory gives the mapping properties for the local sharp maximal functions, the geometric maximal functions and the rough maximal function on the local Morrey spaces with variable exponents. Mathematics subject classification (2010): 42B20, 42B35, 46E30.
TL;DR: The optimal bounds for the arithmetic mean are provided in terms of the arithmetic, harmonic, quadratic, inverse quadRatic and geometric combination of means generated by sine, tangent,hyperbolic sine and hyperbolic tangent functions.
Abstract: We provide the optimal bounds for the arithmetic mean in terms of the arithmetic, harmonic, quadratic, inverse quadratic and geometric combination of means generated by sine, tangent, hyperbolic sine and hyperbolic tangent functions. Mathematics subject classification (2010): 26D15.
TL;DR: In this article, lower bounds for weakly convergent sequence coefficient WCS(X) of a Banach space X, in terms of the James type constant JX,t (τ), the coefficient of weak orthogonality μ(X), and Domı́nguez-Benavides coefficient R(1,X) were established.
Abstract: In this paper, we establish the lower bounds for the weakly convergent sequence coefficient WCS(X) of a Banach space X , in terms of the James type constant JX ,t (τ) , the coefficient of weak orthogonality μ(X) and Domı́nguez-Benavides coefficient R(1,X) . By mean of these bounds, we identify some geometrical properties implying normal structure. Meanwhile, the James type constant JX ,t (τ) , the coefficient of weak orthogonality μ(X) and Domı́nguezBenavides coefficient R(1,X) for the Bynum space l2,∞ are computed to show that our estimates are sharp. Mathematics subject classification (2010): 46B20, 46B25.
TL;DR: In this paper, the sharp factor of the two-sides estimates of the optimal constant in generalized Hardy's inequality with two general Borel measures on $\mathbb{R}$, which generalizes and unifies the known continuous and discrete cases.
Abstract: We obtain the sharp factor of the two-sides estimates of the optimal constant in generalized Hardy's inequality with two general Borel measures on $\mathbb{R}$, which generalizes and unifies the known continuous and discrete cases.
TL;DR: In this article, a new subadditivity behavior of convex and concave functions, when applied to Hilbert space operators, was presented, under suitable assumptions on the spectrum of the positive operators $A$ and $B$.
Abstract: In this article, we present a new subadditivity behavior of convex and concave functions, when applied to Hilbert space operators. For example, under suitable assumptions on the spectrum of the positive operators $A$ and $B$, we prove that \[{{2}^{1-r}}{{\left( A+B \right)}^{r}}\le {{A}^{r}}+{{B}^{r}}\quad\text{ for }r>1\text{ and }r<0,\] and \[{{A}^{r}}+{{B}^{r}}\le {{2}^{1-r}}{{\left( A+B \right)}^{r}}\quad\text{ for }r\in \left[ 0,1 \right].\] These results provide considerable generalization of earlier results by Aujla and Silva.
Further, we present several extensions of the subadditivity idea initiated by Ando and Zhan, then extended by Bourin and Uchiyama.
TL;DR: In this paper, the authors generalize several results related to the Bohr radius for analytic functions or harmonic functions on the unit disc U in C to holomorphic mappings or pluriharmonic mappings on BX.
Abstract: Let BX be the unit ball of a complex Banach space X . In this paper, we will generalize several results related to the Bohr radius for analytic functions or harmonic functions on the unit disc U in C to holomorphic mappings or pluriharmonic mappings on BX . We will establish Bohr’s inequality for the class of holomorphic mappings which are subordinate to convex mappings on BX . Next, we will establish Bohr’s inequality for pluriharmonic mappings on BX . We will also obtain the p -Bohr radius for bounded pluriharmonic functions on BX . Finally, we will determine the Bohr radius for a class of holomorphic functions on BX which contains odd holomorphic functions on BX . Mathematics subject classification (2010): 32A05, 32A10, 32K05.
TL;DR: Rearrangements and Jensen type inequalities related to convexity, super-quadracity, strongconvexity and 1-quasiconvexities are given in this article.
Abstract: Rearrangements andJensen type inequalities related to convexity, superquadracity, strongconvexity and 1-quasiconvexity
TL;DR: In this paper, a generalization to matrices and tensors of the Szőkefalvi-Nagy inequality and the Grüss-Popoviciu inequality is presented.
Abstract: We prove a generalization to matrices and tensors of the Szőkefalvi-Nagy inequality and the Grüss-Popoviciu inequality. Our more general version is required in the analysis of variance (ANOVA).
TL;DR: In this paper, connections between m -PPT maps, m -positive maps and m -copositive maps are given, and characterizations of completely PPT maps are obtained.
Abstract: Linear maps Φ : Mn → Mk are called m -PPT if [Φ(Ai j)]i, j=1 are positive partial transpose matrices for all positive semi-definite matrices [Ai j]i, j=1 ∈ Mm(Mn) . In this paper, connections between m -PPT maps, m -positive maps and m -copositive maps are given. In consequence, characterizations of completely PPT maps are obtained. The results are applied to study two linear maps X → X + a(trX)I and X → a(trX)I−X for a 0 . Moreover, singular values inequalities of 2× 2 positive block matrices under these two linear maps are given. In particular, we prove an open singular values inequality formulated by Lin [Linear Algebra Appl, 520 (2017)] for n 3. Mathematics subject classification (2010): 15A18, 15A45.