Showing papers in "Mathematical Inequalities & Applications in 2015"
TL;DR: In this article, Lyapunov-type inequalities are established for a fractional differential equation under mixed boundary conditions, and intervals where certain MittagLeffler functions have no real zeros are obtained.
Abstract: Lyapunov-type inequalities are established for a fractional differential equation under mixed boundary conditions. Using such inequalities, we obtain intervals where certain MittagLeffler functions have no real zeros. Mathematics subject classification (2010): 34A08, 34A40, 26D10, 33E12.
78 citations
TL;DR: In this paper, the authors derived several Hilbert-type inequalities with a differential operator regarding a general homogeneous kernel, and showed that the constants appearing on the right-hand sides of these inequalities are the best possible.
Abstract: Motivated by some recent results, in this article we derive several Hilbert-type inequalities with a differential operator, regarding a general homogeneous kernel. Moreover, we show that the constants appearing on the right-hand sides of these inequalities are the best possible. The general results are then applied to some particular examples of homogeneous kernels and compared with previously known from the literature. Mathematics subject classification (2010): Primary 26D10, 26D15, Secondary 33B15.
40 citations
TL;DR: In this paper, the authors established inequalities, monotonicity, convexity, and unimodality for functions concerning the modified Bessel functions of the first kind and computed the completely monotonic degrees of differences between the exponential and trigamma functions.
Abstract: In the paper, the author establishes inequalities, monotonicity, convexity, and unimodality for functions concerning the modified Bessel functions of the first kind and compute the completely monotonic degrees of differences between the exponential and trigamma functions. Mathematics subject classification (2010): Primary 26A12; Secondary 26A48, 26A51, 26D15, 30D10, 30E20, 33B10, 33B15, 33C20, 42B10, 44A10.
35 citations
TL;DR: In this paper, a more accurate multidimensional discrete Hilbert-type inequality with a best possible constant factor and some parameters is given by using the way of weight coefficients and technique of real analysis and complex analysis.
Abstract: In this paper, by using the way of weight coefficients and technique of real analysis and complex analysis, a more accurate multidimensional discrete Hilbert-type inequality with a best possible constant factor and some parameters is given. The equivalent form, the operator expression with the norm are also considered. Mathematics subject classification (2010): 26D15, 47A07.
26 citations
TL;DR: In this paper, the necessary and sufficient conditions on inear operators A and B for inequality of a function f that is n-convex at a point were studied for every function f.
Abstract: We study necessary and sufficient conditions on inear operators A and B for inequality Af <= Bf to hold for every function f that is n-convex at a point.
25 citations
TL;DR: In this article, the strong subadditivity of the von Neumnann entropy can be derived from the monotonicity of the Umegaki relative entropy, which is an inequality of relative quasi-entropies.
Abstract: In this paper we investigate the inequality Sq(ρ123)+ Sq(ρ2) Sq(ρ12)+ Sq(ρ23)(∗) where ρ123 is a state on a finite dimensional Hilbert space H1⊗H2⊗H3, and Sq is the Tsallis entropy. It is well-known that the strong subadditivity of the von Neumnann entropy can be derived from the monotonicity of the Umegaki relative entropy. Now, we present an equivalent form of (*), which is an inequality of relative quasi-entropies. We derive an inequality of the form Sq(ρ123)+Sq(ρ2) Sq(ρ12)+Sq(ρ23)+ fq(ρ123) , where f1(ρ123) = 0 . Such a result can be considered as a generalization of the strong subadditivity of the von Neumnann entropy. One can see that (*) does not hold in general (a picturesque example is included in this paper), but we give a sufficient condition for this inequality, as well. Mathematics subject classification (2010): 46N50, 94A15, 46L30, 47L90.
25 citations
TL;DR: The notion of Lp-centroid bodies was introduced by Wang, Lu and Leng as discussed by the authors, and the extremal values of dual quermassintegrals of the polars of general Lp centroid bodies are also provided.
Abstract: In this article, we define the general Lp -centroid bodies, which extend the notion of Lp -centroid bodies by Lutwak and Zhang. Further, we generalize the two monotone inequalities by Wang, Lu and Leng, and establish the Brunn-Minkowski type inequalities of dual quermassintegrals for this new notion. In particular, the extremal values of dual quermassintegrals of the polars of general Lp -centroid bodies are also provided. Mathematics subject classification (2010): 52A20, 52A40.
21 citations
TL;DR: In this paper, by using the concept of Orlicz mixed volume, the authors extend geominimal surface area to the ORlicz version and give some properties and an isoperimetric inequalities for the ORL.
Abstract: In 1996, E. Lutwak extended the important concept of geominimal surface area to Lp version, which serves as a bridge connecting a number of areas of geometry: affine differential geometry, relative differential geometry, and Minkowskian geometry. In this paper, by using the concept of Orlicz mixed volume, we extend geominimal surface area to the Orlicz version and give some properties and an isoperimetric inequalities for the Orlicz geominimal surface areas. Mathematics subject classification (2010): 52A39, 52A40.
21 citations
TL;DR: In this paper, sufficient conditions for the boundedness of fractional Hausdorff operators on the Lebesgue spaces with power weights with respect to some special cases were given, and a better lower bound was obtained compared with a result of the paper.
Abstract: In this paper, we give the sufficient conditions for the boundedness of the (fractional) Hausdorff operators on the Lebesgue spaces with power weights. In some special cases, these conditions are the same and also necessary. As an application, we obtain a better lower bound of fractional Hardy operators on the Lebesgue spaces compared with a result of the paper [25]. Mathematics subject classification (2010): 26D10, 26D15, 42B35, 46E30.
20 citations
TL;DR: In this paper, it was shown that Taylor's polynomials for logF provide upper and lower bounds for log F according to the parity of their degree, and the formula connecting the Beta function to the Gamma function shows that the bounds for F are actually bounds for Beta, and that when k is even the conclusion holds for every X,Y ∈ R with (X,Y) = (0,0).
Abstract: Let g(x):= (e/x)xΓ(x+1) and F(x,y):= g(x)g(y)/g(x+y). Let Dx,y (k) be the k th differential in Taylor's expansion of logF(x,y) . We prove that when (x,y) ∈ R+ 2 one has (-1)k-1Dx,y (k) (X,Y) > 0 for every X,Y ∈ R+, and that when k is even the conclusion holds for every X,Y ∈ R with (X,Y) = (0,0). This implies that Taylor's polynomials for logF provide upper and lower bounds for logF according to the parity of their degree. The formula connecting the Beta function to the Gamma function shows that the bounds for F are actually bounds for Beta.
20 citations
TL;DR: In this article, the Hermite-Hadamard-Fejer type inequalities for delta-convex functions of higher order were investigated and the results were applied to derive some inequalities between quadrature operators.
Abstract: In our previous paper [15], using s -convex stochastic ordering [4], we investigate Hermite-Hadamard-Fejer type inequalities in the case of higher order convex functions. In the present paper, our aim is to extend this investigation from convex to delta-convex functions of higher order [8]. We offer some useful tools for obtaining and proving of various forms of the Hermite-Hadamard-Fejer type inequalities for delta-convex functions of higher order, that generalizes results of Dragomir et al. [5]. These results are applied to derive some inequalities between quadrature operators. We define also and study strong delta-convexity of n -th order that generalizes strong n -convexity studied in [14] and [9]. Mathematics subject classification (2010): 26A51, 39B62.
TL;DR: In this paper, the parametric Marcinkiewicz integrals with mixed homogeneity along certain compound surfaces were considered and the Lp boundedness of the integral kernels was shown for both on the unit sphere and in the radial direction.
Abstract: In this paper we consider the parametric Marcinkiewicz integrals with mixed homogeneity along certain compound surfaces. Under the rather weakened size conditions on the integral kernels both on the unit sphere and in the radial direction, the Lp boundedness for such operators are given. As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood-Paley gλ functions are also obtained. Mathematics subject classification (2010): 42B20, 42B25, 42B99.
TL;DR: In this paper, the generalized Popoviciu inequality for convex functions is generalized for higher order convex function functions via generalized Montgomery identity, and the bounds for the identities related to the generalization of the Popovichiu inequality using inequalities for the Cebysev functional are constructed.
Abstract: We obtained useful identities via generalized Montgomery identity, by which the inequality of Popoviciu for convex functions is generalized for higher order convex functions. We investigate the bounds for the identities related to the generalization of the Popoviciu inequality using inequalities for the Cebysev functional. Some results relating to the Gruss and Ostrowski type inequalities are constructed. Further, we also construct new families of exponentially convex functions and Cauchy-type means by looking at linear functionals associated with the obtained inequalities.
TL;DR: In this paper, the boundedness of Volterra type operators on Morrey type spaces was investigated and a boundedness analysis of the VOLTERRA type operator on H2 K was performed.
Abstract: In this paper, we investigate the boundedness of Volterra type operators on Morrey type spaces H2 K . Mathematics subject classification (2010): 30H25, 47B38.
TL;DR: In this paper, the authors used the Holder inequality and Keller's chain rule on a time scale T = N to prove the discrete inequalities due to Bennett and Leindler which are converses of Copson's inequalities.
Abstract: In this paper, we will prove some new dynamic inequalities on a time scale T . These inequalities when T = N contain the discrete inequalities due to Bennett and Leindler which are converses of Copson’s inequalities. The main results will be proved using the Holder inequality and Keller’s chain rule on time scales. Mathematics subject classification (2010): 26A15, 26D10, 26D15, 39A13, 34A40. 34N05.
TL;DR: In this paper, the authors determined the ratio of the normalized Lommel functions Lμ,ν of the form (??) to its sequence of partial sums (Lμ,ν ) m (z) = z + m ∑ n=1 anz when the coefficients of Lμ andν satisfy some conditions, and investigated the radii of univalency, starlikeness, convexity and close-to-convexity of the partial sums.
Abstract: The aim of the present paper determine the ratio of the normalized Lommel functions Lμ ,ν of the form (??) to its sequence of partial sums ( Lμ ,ν ) m (z) = z + m ∑ n=1 anz when the coefficients of Lμ ,ν satisfy some conditions. Furthermore we investigate the radii of univalency, starlikeness, convexity and close-to-convexity of the partial sums ( Lμ ,ν ) m (z) . Computational and graphical usages of Maple (Version 17) as well as geometrical descriptions of the image domains in several illustrative examples are also presented. Mathematics subject classification (2010): 30C45, 33C10.
TL;DR: In this article, some operator inequalities related to Tsallis relative operator entropy are presented, which are refinements and generalizations of some existing inequalities, and are used for subject classification.
Abstract: In this paper, we present some operator inequalities related to Tsallis relative operator entropy. Our results are refinements and generalizations of some existing inequalities. Mathematics subject classification (2010): 47A63.
TL;DR: In this article, the authors proved diamond-alpha dynamic inequalities of Opial type with one and two weight functions on time scales, which contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case.
Abstract: In this paper, we prove some new diamond-alpha dynamic inequalities of Opial type with one and with two weight functions on time scales. These results contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case. Mathematics subject classification (2010): 39A10, 39A12, 26D15.
TL;DR: In this paper, the authors get to the bottom of the origin of Polya's integral inequality, plots out the development of the theory of inequalities, collects variants and proofs of PolyA's integral inequalities, surveys Iyengar-Mahajani's, Agarwal-Dragomir's, Cerone-Dragoma's, and Qi's refinements, generalizations, and applications of PolyAs integral inequalities and find equivalences between these integral inequalities.
Abstract: In the article, the author gets to the bottom of the origin of Polya’s integral inequality, plots out the development of the theory of inequalities, collects variants and proofs of Polya’s integral inequality, surveys Iyengar-Mahajani’s, Agarwal-Dragomir’s, Cerone-Dragomir’s, and Qi’s refinements, generalizations, and applications of Polya’s integral inequality, and find equivalences between these integral inequalities.
TL;DR: In this article, the operator norm of the fractional Hardy operator Hβ from L p(Rn) to L q (Rn), where 0 < β < n, 1 < p < q < ∞ and 1/p−1/q = β/n, was precisely evaluated.
Abstract: In this note, we precisely evaluate the operator norm of the fractional Hardy operator Hβ from L p(Rn) to Lq(Rn) , where 0 < β < n , 1 < p < q <∞ and 1/p−1/q = β/n . By this we extend the result of Bliss [1] to the case of high dimension and improve our result in [7]. Mathematics subject classification (2010): Primary 26D10, 26D15, 42B99.
TL;DR: In this article, the authors construct the weight wr( ; ) defined on QQ and extension operator Ext L : Lipd(¶ Q)7! Lip(Q) from Lipschitz functions defined onQQ with certain restricted support to Lipshitz functions on Q, independent of r and R, in such a way that Ext L extends to the bounded operator from certain subspace of weighted Orlicz-Slobodetski space Y R;R wr (¶ Q), subordinated to the weight r to Orliczi Sobolev space W
Abstract: Having given weight ˜ r = r(dist(x;¶ Q)) defined on cube Q and Orlicz function R , we construct the weight wr( ; ) defined onQQ and extension operator Ext L : Lipd(¶ Q)7! Lip(Q) from Lipschitz functions defined onQ with certain restricted support to Lipschitz functions defined on Q , independent of r and R , in such a way that Ext L extends to the bounded operator from certain subspace of weighted Orlicz-Slobodetski space Y R;R wr (¶ Q) subordinated to the weight wr to Orlicz Sobolev space W 1;R r (Q) . Result is new in the unweighted Orlicz setting for general function R as well as in the weighted L p setting.
TL;DR: In this article, a vector majorization ordering for comparing two m -tuples of vectors of a real linear space was proposed, which extends the classical approach of scalar majorization theory for comparing m − tuples of scalars in R.
Abstract: In this paper we study a vector majorization ordering for comparing two m -tuples of vectors of a real linear space. This extends the classical approach of (scalar) majorization theory for comparing m -tuples of scalars in R . We prove a Sherman type inequality for a vectorvalued C -convex function f , where C is a cone ordering. In consequence, we obtain a Hardy-Littlewood-Pólya-Karamata type inequality generated by m -tuples of vectors in a vector space. As applications, we present majorization generalizations of the superadditivity properties of the Jensen and Jensen-Mercer functionals generated by a convex function f . In addition, we show that some sums generated by the Jensen and Jensen-Mercer functionals are Schur-concave with respect to their weight vectors. We also give interpretations of the obtained results for tridiagonal doubly stochastic matrices and doubly stochastic circular matrices. Mathematics subject classification (2010): 52A40, 06F20, 26B25, 15A39.
TL;DR: In this paper, the authors generalize Steffensen's inequality for positive measures and obtain conditions for these inequalities which are invariant in form to Steffensens inequality for absolutely continuous measures.
Abstract: We generalize Steffensen’s inequality for positive measures. We obtain conditions for these inequalities which are invariant in form to Steffensen’s inequality for absolutely continuous measures. Further, we produce linear functionals which generate exponential convexity and Cauchy means.
TL;DR: In this paper, a new criterion for boundedness and compactness of generalized weighted composition operators from H ∞ into the Zygmund space is proposed, based on the notion of boundedness.
Abstract: A new criterion for the boundedness, as well as for the compactness of the generalized weighted composition operators from H∞ into the Zygmund space are given in this paper. Mathematics subject classification (2010): 47B33, 30H30.
TL;DR: In this paper, some new dynamic inequalities with two weight functions and two unknown functions of Opial type on time scales were proved by employing Hölder's inequality, the chain rule and some basic algebraic inequalities.
Abstract: In this paper we prove some new dynamic inequalities with two weight functions and some new dynamic inequalities with two unknown functions of Opial type on time scales. The main results will be proved by employing Hölder’s inequality, the chain rule and some basic algebraic inequalities. Mathematics subject classification (2010): 26A15, 26D10, 26D15, 39A13, 34A40.
TL;DR: In this paper, it is proved that det (In +T ∗T ) det(Ir +X∗X) ·det (In−r +Z∗Z) with equality if and only if Y = 0.
Abstract: Let T = [ X Y 0 Z ] be an n -square matrix, where X ,Z are r -square and (n− r) -square, respectively. Among other determinantal inequalities, it is proved that det (In +T ∗T ) det(Ir +X∗X) ·det (In−r +Z∗Z) with equality if and only if Y = 0 . Mathematics subject classification (2010): 15A45.
TL;DR: In this paper, the authors presented an estimate for the distance among quasi-arithmetic means whose underlying functions satisfy some smoothness conditions, by using Pales' operator and by using Mikusinski and Łojasiewicz's operator.
Abstract: In the 1960s Cargo and Shisha proved some majorizations for the distance among quasi-arithmetic means (defined as f−1 (∑i=1 wi f (ai)) for any continuous, strictly monotone function f : I → R , where I is an interval, and (a1, . . . ,an) is a vector with entries in I , (w1, . . . ,wn) is a sequence of corresponding weights wi > 0 , ∑wi = 1). Nearly thirty years later, in 1991, Pales presented an iff condition for a sequence of quasi-arithmetic means to converge to another QA mean. It was closely related with the threeparameter operator ( f (x)− f (y))/( f (x)− f (z)) . The author presented recently an estimate for the distance among such quasi-arithmetic means whose underlying functions satisfy some smoothness conditions. Used was the operator f → f ′′/ f ′ introduced in the 1940s by Mikusinski and Łojasiewicz. It is natural to look for similar estimate(s) in the case of the underlying functions not being smooth. For instance, by the way of using Pales’ operator. This is done in the present note. Moreover, the result strengthens author’s earlier estimates. Mathematics subject classification (2010): 26E60, 26D15, 26D07.
TL;DR: In this article, the weighted approximation error of Baskakov operator was characterized for weights of the form w(x) = xγ0 (1+ x)γ∞, where γ0 ∈ [−1,0], γ∞ ∈ R.
Abstract: The weighted approximation errors of Baskakov operator is characterized for weights of the form w(x) = xγ0 (1+ x)γ∞ , where γ0 ∈ [−1,0] , γ∞ ∈ R . Direct inequalities and strong converse inequalities of type A are proved in terms of the weighted K -functional. Mathematics subject classification (2010): 41A36, 41A25, 41A27, 41A17.
TL;DR: In this article, the authors studied the functorial properties of the spaces R(X), which have been recently introduced as a central tool in the analysis of the Hardy operator minus the identity on decreasing functions.
Abstract: We study functorial properties of the spaces R(X) , which have been recently introduced as a central tool in the analysis of the Hardy operator minus the identity on decreasing functions. In particular, we provide conditions on a minimal Lorentz space Λφ so that the equation R(X) = Λφ has a solution within the category of rearrangement invariant (r.i.) spaces. Moreover, we show that if R(X) =Λφ , then we can always take X to be the minimal r.i. Banach range space for the Hardy operator defined in Λφ . Mathematics subject classification (2010): 26D10, 46E30.
TL;DR: In this paper, the authors prove two conjectures of Chen concerning the complete monotonicity properties of some functions involving the gamma and polygamma functions, and derive recursive relations of the Bernoulli numbers.
Abstract: We prove two conjectures of Chen concerning the complete monotonicity properties of some functions involving the gamma and polygamma functions. We prove asymptotic expansions of the logarithm of the gamma function in terms of the polygamma functions, and provide recurrence relations to calculate the coefficients of the asymptotic expansions. By using the results obtained, we derive recursive relations of the Bernoulli numbers. Mathematics subject classification (2010): 33B15, 26A48, 41A60.