Showing papers in "Mathematical Inequalities & Applications in 2014"
TL;DR: In this paper, the boundedness of generalized fractional integral operators on Morrey spaces has been studied and three necessary and sufficient conditions for their boundedness were proved. But these conditions are independent.
Abstract: We present some characterizations for the boundedness of the generalized fractional integral operators on Morrey spaces. The characterizations follow from two key estimates, one for the norm of some functions in Morrey spaces, and another for the values of the corresponding fractional integrals. We prove three theorems about necessary and sufficient conditions. We show that these theorems are independent by giving some examples. We also obtain counterparts for
the weak generalized Morrey spaces.
46 citations
TL;DR: Using Ohlin's Lemma on convex stochastic ordering, the authors gave a simple proof of known Hermite-Hadamard-Fejer type inequalities, and also proved new inequalities.
Abstract: Using Ohlin’s Lemma [21] on convex stochastic ordering, we get a simple proof of known Hermite-Hadamard-Fejer type inequalities. We also prove new inequalities. Using s convex stochastic ordering [12], we also give some Hermite-Hadamard-Fejer type inequalities in the case of higher order convex functions. The obtained results are useful in proving some inequalities between the quadrature operators [31], [32]. Mathematics subject classification (2010): 26A51, 26D15, 60E15, 41A55.
24 citations
TL;DR: In this article, the authors established, by Cauchy integral formula in the theory of complex functions, Levy-Khintchine representation for the geometric mean of many positive numbers, and found that the geometric means of many numbers is a complete Bernstein function, and provided a new proof of the well known arithmetic-geometric mean inequality.
Abstract: In the paper, the authors establish, by Cauchy integral formula in the theory of complex functions, Levy-Khintchine representation for the geometric mean of many positive numbers, find that the geometric mean of many positive numbers is a complete Bernstein function, and supply a new proof of the well known arithmetic-geometric mean inequality. Mathematics subject classification (2010): Primary 26E60; Secondary 26A48, 30E20, 44A10, 44A20.
23 citations
TL;DR: In this paper, the left-star and right-star partial orders on B(H) are given and bijective additive maps on H which preserve the leftstar or the rightstar partial order in both directions are characterized.
Abstract: Let H be an infinite dimensional complex Hilbert space, and let B(H) be the set of all bounded linear operators on H . In the paper equivalent definitions for the left-star and the right-star partial orders on B(H) are given and bijective additive maps on B(H) which preserve the left-star or the right-star partial order in both directions are characterized. Mathematics subject classification (2010): 06A06, 15A03, 15A04, 15A86.
21 citations
TL;DR: In this article, the authors presented new sharp bounds for Seiffert P in terms of weighted power means of arithmetic mean A and geometric mean G : (2 3 A p1 + 3 G p1 )1/p1 < P < ( 2 3 Ap2 + 3G p2 )1 /p2, where p1 = 4/5 and p2 = logπ/2 (3/2) are the best possible constants.
Abstract: For a,b > 0 with a = b , let P = (a− b)/(4arctana/b−π) , A = (a+ b)/2 , G = √ ab denote the Seiffert mean, arithmetic mean, geometric mean of a and b , respectively. In this paper, we present new sharp bounds for Seiffert P in terms of weighted power means of arithmetic mean A and geometric mean G : ( 2 3 A p1 + 3 G p1 )1/p1 < P < ( 2 3 A p2 + 3 G p2 )1/p2 , where p1 = 4/5 and p2 = logπ/2 (3/2) are the best possible constants. Moreover, our sharp bounds for P are compared with other known ones, which yields a chain of inequalities involving Seiffert mean P . Mathematics subject classification (2010): Primary 26E60, 26D05; secondary 33B10.
18 citations
TL;DR: In this article, the sufficient and necessary conditions for the boundedness of Hausdorff operators on various function spaces were given, and the Lipschitz estimates for the commutator of the operator were considered.
Abstract: In this paper, we give the sufficient and necessary conditions for the boundedness of Hausdorff operators on various function spaces. Moreover, we consider Lipschitz estimates for the commutator of Hausdorff operators. We extend some known results. Mathematics subject classification (2010): 26D10, 26D15, 46E30.
17 citations
TL;DR: In this article, the Stirling formula for the q gamma function is exploited to prove the complete monotonicity property of functions involving the q -gamma and the q-digamma functions.
Abstract: In this paper, the q -analogue of the Stirling formula (the Moak formula) for the q gamma function is exploited to prove the complete monotonicity property of functions involving the q -gamma and the q -digamma functions. The monotonicity of these functions is used to establish sharp inequalities for the q -gamma and the q -polygamma functions and the q -Harmonic number. Mathematics subject classification (2010): 33D05, 26D07, 26A48.
17 citations
TL;DR: In this paper, the Fourier multiplier theorems on Triebel-Lizorkin spaces with variable exponents were shown to coincide with variable Bessel potential spaces, variable Sobolev spaces and variable Lebesgue spaces when appropriate indices are chosen.
Abstract: In this paper, we will prove Fourier multiplier theorems on Besov and Triebel–Lizorkin spaces with variable exponents. It was shown by many authors that variable Triebel–Lizorkin spaces coincide with variable Bessel potential spaces, variable Sobolev spaces and variable Lebesgue spaces when appropriate indices are chosen. In consequence of the results, we also have Fourier multiplier theorems on these variable function spaces. Mathematics subject classification (2010): 42B15.
16 citations
TL;DR: In this article, two new forms of the Hilbert inequality are introduced, the first is a sharper form of the classical Hilbert inequality and is connected to Hardy inequality, and the second one is a differential form of Hilbert inequality.
Abstract: In this paper we obtain two new forms of the Hilbert inequality. The first form is a sharper form of the classical Hilbert inequality and is connected to Hardy inequality. In the second one we introduce a differential form of Hilbert inequality. Mathematics subject classification (2010): 26D15.
15 citations
TL;DR: In this article, it was shown that finite-dimensional univariate function spaces satisfying a Bernstein-like inequality admit norming meshes, in particular meshes with "optimal" cardinality for trigonometric polynomials on subintervals of the period.
Abstract: We show that finite-dimensional univariate function spaces satisfying a Bernstein-like inequality admit norming meshes. In particular, we determine meshes with “optimal” cardinality for trigonometric polynomials on subintervals of the period. As an application we discuss the construction of optimal bivariate polynomial meshes by arc blending.
13 citations
TL;DR: In this paper, the boundedness and compactness of the products of the radial derivative and multiplication operator RMu between mixed norm spaces H(p, q, φ) and Zygmund-type spaces on the unit ball were characterized.
Abstract: In this paper, we obtain some characterizations of the boundedness and compactness of the products of the radial derivative and multiplication operator RMu between mixed norm spaces H(p, q, φ) and Zygmund-type spaces on the unit ball.
TL;DR: In this article, Sherman's inequality is extended from convex functions to the class of (α, β) -convex functions including (k,h) -Convex Functions.
Abstract: In this paper, Sherman’s inequality is extended from convex functions to the class of (α ,β) -convex functions including (k,h) -convex functions. Sherman’s type results corresponding to Jensen-Steffensen, Mercer-Steffensen and Brunk inequalities are established. The obtained results are applied to mixed (α ,β) -convex functions. Mathematics subject classification (2010): 39B62, 26D15.
TL;DR: In this article, weakly equilibrium Cantor-type sets are used to solve two problems related to polynomial inequalities: the problem by M. Baran et al. about a compact set K ⊂ C such that the Markov inequality is not valid on K with the best Markov's exponent, and the problem concerning compact sets satisfying the local form of Markov’s inequality with a given exponent, but not satisfying the global version of markov's inequality with the same parameter.
Abstract: By means of weakly equilibrium Cantor-type sets, solutions of two problems related to polynomial inequalities are presented: the problem by M. Baran et al. about a compact set K ⊂ C such that the Markov inequality is not valid on K with the best Markov’s exponent, and the problem by L. Frerick et al. concerning compact sets satisfying the local form of Markov’s inequality with a given exponent, but not satisfying the global version of Markov’s inequality with the same parameter. Mathematics subject classification (2010): 41A17, 41A44.
TL;DR: In this paper, the uniqueness of the solution of the initial value problem involving higher order partial differential equation is proved. But the problem is not solved in this paper, since it is not a special case of the Opial-type inequalities.
Abstract: In the present paper we establish some new Opial-type inequalities involving higher order partial derivatives. Our results in special cases yield some of the recent results on Opial’s inequality. As application, we prove the uniqueness of the solution of initial value problem involving higher order partial differential equation. Mathematics subject classification (2010): 26D15.
TL;DR: In this paper, the analysis of bivariate parameter means: general power mean, generalized logarithmic mean, Gini mean, Stolarsky mean and Gini coefficient is presented.
Abstract: C Abstract. The subject of this paper is the analysis of bivariate parameter means: general power mean, generalized logarithmic mean, Gini mean and Stolarsky mean. Asymptotical analysis of these means are made and series of corresponding coefficients are calculated. Using these in- formation, a necessary conditions for the comparison of these means are derived. This approach enables better understanding of relations between these means.
TL;DR: In this paper, the decay rates for integral operators generated by power series-like kernels on a subset X of either Rq or Cq were derived from decay assumptions on the sequence of coefficients in the expansion of the kernel and on the orthogonal family.
Abstract: We deduce decay rates for eigenvalues of integral operators generated by power serieslike kernels on a subset X of either Rq or Cq . A power series-like kernel is a Mercer kernel having a series expansion based on an orthogonal family { fα}α∈Zq+ in L 2(X ,μ) , in which μ is a complete measure on X . As so, we show that the eigenvalues of the integral operators are given by an explicit formula defined by the coefficients in the series expansion of the kernel and the elements of the orthogonal family. The inequalities and, in particular, the decay rates for the sequence of eigenvalues are obtained from decay assumptions on the sequence of coefficients in the expansion of the kernel and on the sequence {‖ fα‖}α∈Zq+ . Mathematics subject classification (2010): 45P05, 32A05, 47B65, 42C05.
TL;DR: In this paper, the authors investigated some problems related with Berezin symbols of operators on Hardy and Bergman spaces and their applications in summability theory and in solution of Beurling problem.
Abstract: We investigate some problems related with Berezin symbols of operators on Hardy and Bergman spaces and their applications in summability theory and in solution of Beurling problem. We also study boundedness and invertibility of some Toeplitz products on the Hardy and Bergman spaces. Mathematics subject classification (2010): Primary 47B35.
TL;DR: In this article, the Hardy-Littlewood maximal function is characterized on the Lebesgue spaces with variable exponent, where the commutator generated by the function and a suitable function b are defined by [M,b] f = M(b f )−bM f.
Abstract: Let M be the Hardy-Littlewood maximal function, the commutator generated by M and a suitable function b is defined by [M,b] f = M(b f )−bM f . In this paper, the authors give some characterizations of b for which [M,b] is bounded on the Lebesgue spaces with variable exponent. The similar results are also proved for the commutator of the sharp maximal function. Mathematics subject classification (2010): 42B25, 46E30.
TL;DR: In this article, it was shown that the generalized local Morrey spaces are embedded between weighted Lebesgue spaces with weights differing only by a logarithmic factor, which leads to the statement that these embeddings are strict.
Abstract: We prove a new property of Morrey function spaces by showing that the generalized local Morrey spaces are embedded between weighted Lebesgue spaces with weights differing only by a logarithmic factor. This leads to the statement that the generalized global Morrey spaces are embedded between two generalized Stummel classes whose characteristics similarly differ by a logarithmic factor. We give examples proving that these embeddings are strict. For the generalized Stummel spaces we also give an equivalent norm.
TL;DR: In this article, the authors prove monotonicity and convexity results for the modified Struve function of the second kind by using its integral representation, and present some functional inequalities (like Tur´an type inequalities) and lower and upper bounds for modified Struve function and its logarithmic derivative.
Abstract: In this paper our aim is to prove some monotonicity and convexity results for the
modified Struve function of the second kind by using its integral representation. Moreover, as
consequences of these results, we present some functional inequalities (like Tur´an type inequalities)
as well as lower and upper bounds for modified Struve function of the second kind and its
logarithmic derivative.
TL;DR: In this paper, the authors established several new Lyapunov-type inequalities for two classes of Dirichlet quasilinear systems, which almost generalize and extend all related existing results in the literature.
Abstract: In this paper, we establish several new Lyapunov-type inequalities for two classes of Dirichlet quasilinear systems, which almost generalize and extend all related existing results in the literature. As an application, we also obtain sharp lower bounds for the eigenvalues of corresponding systems. Mathematics subject classification (2010): 26D10, 34A40, 34C10.
TL;DR: In this article, the authors investigated the boundedness of a general convolution operator between certain weighted Lorentz-type spaces with the aim of proving analogues of the Young convolution inequality for these spaces.
Abstract: This thesis is devoted to an investigation of boundedness of a general convolution operator between certain weighted Lorentz-type spaces with the aim of proving analogues of the Young convolution inequality for these spaces.Necessary and sufficient conditions on the kernel function are given, for which the convolution operator with the fixed kernel is bounded between a certain domain space and the weighted Lorentz space of type Gamma. The considered domain spaces are the weighted Lorentz-type spaces defined in terms of the nondecreasing rearrangement of a function, the maximal function or the difference of these two quantities.In each case of the domain space, the corresponding Young-type convolution inequality is proved and the optimality of involved rearrangement-invariant spaces in shown.Furthermore, covering of the previously existing results is also discussed and some properties of the new rearrangement-invariant function spaces obtained during the process are studied.
TL;DR: In this article, it was proved that the boundedness of the maximal operator M from a Lebesgue space Lp1(R n) to a general local Morrey-type space LMp2θ,w(r n) for any 0 1 is equivalent to boundedness for the Hardy operator from L p1 p2 (0, ∞) to the weighted Lebesgege space L θ p2,v(0,∞) for a certain weight function v determined by the functional parameter w.
Abstract: It is proved that the boundedness of the maximal operator M from a Lebesgue space Lp1(R n) to a general local Morrey-type space LMp2θ ,w(R n) is equivalent to the boundedness of the embedding operator from Lp1 (R n) to LMp2θ ,w(R n) and in its turn to the boundedness of the Hardy operator from L p1 p2 (0,∞) to the weighted Lebesgue space L θ p2 ,v(0,∞) for a certain weight function v determined by the functional parameter w . This allows obtaining necessary and sufficient conditions on the function w ensuring the boundedness of M from Lp1(R n) to LMp2θ ,w(R n) for any 0 1 . These conditions with p1 = p2 = 1 are necessary and sufficient for the boundedness of M from L1(R) to the weak local Morreytype space WLM1θ ,w(R) . Mathematics subject classification (2010): 42B20, 42B25, 42B35.
TL;DR: In this article, several bounds for the perimeter of an ellipse in terms of arithmetic, geometric, and harmonic means are presented, which improve some known results. But they do not consider the geometric properties of the ellipses.
Abstract: In this paper, we present several bounds for the perimeter of an ellipse in terms of arithmetic, geometric, and harmonic means, which improve some known results. Mathematics subject classification (2010): 41A10, 33E05, 33C05, 26E60.
TL;DR: In this article, the main results were proved using algebraic inequalities, Holder inequality, and Keller's chain rule on time scales, and the main result was proved using the algebraic inequality and the chain rule.
Abstract: In this paper we prove some new dynamic inequalities of Hardy type on time scales. The main results will be proved using algebraic inequalities, Holder inequality and Keller’s chain rule on time scales. Mathematics subject classification (2010): 26A15, 26D10, 26D15, 39A13, 34A40, 34N05.
TL;DR: In this article, the authors define new Euler sequence spaces and construct Schauder basis of these spaces, and give the characterization of some classes of compact operators on these spaces by using the Hausdorff measure of noncompactness.
Abstract: In this paper, we define some new Euler sequence spaces and construct Schauder basis of these spaces. Moreover, we determine their β−duals and characterize some related matrix classes. Finally, we give the characterization of some classes of compact operators on these spaces by using the Hausdorff measure of noncompactness. Mathematics subject classification (2010): 46B45, 46B15, 46B50.
TL;DR: In this paper, a more accurate half-discrete Hilbert-type inequality with a general non-homogeneous kernel and a best possible constant factor is given by using the methods of weight functions and technique of real analysis.
Abstract: In this paper, by the use of the methods of weight functions and technique of real analysis, a more accurate half-discrete Hilbert-type inequality with a general non-homogeneous kernel and a best possible constant factor is given. The equivalent forms and some reverses are obtained. We also consider the operator expressions with the norm and some particular examples. Mathematics subject classification (2010): 26D15, 47A07.
TL;DR: In this article, the authors discuss the extension of inequality R_A >= c/a * r_b + b/a* r_c to the plane of triangle ABC.
Abstract: We discuss the extension of inequality R_A >= c/a * r_b + b/a * r_c to the plane of triangle ABC. Based on the obtained extension, in regard to all three vertices of the triangle, we get the extension of Erdos-Mordell inequality, and some inequalities of Erdos-Mordell type.
TL;DR: In this paper, monotonicity results concerning the gamma function are deduced, which lead to inequalities which improve some known bounds for the Γ function and improve the known bounds.
Abstract: In this paper monotonicity results concerning the gamma function are deduced. These results lead to inequalities which improve some known bounds for the Γ function. Mathematics subject classification (2010): 33B15.
TL;DR: Algebraic approximations for the arctangent functions are presented and it is proved that for instance π(3+8 √ 2)x 7+6 √ 1+ x2 +16√ 2 √1+ x 2 +√ 1- x2 arctan x 45x 7-6 £1.
Abstract: A method of producing refinements of the Shafer-Fink ([5]) inequality 3x 1+2 √ 1+ x2 arctan x πx 1+2 √ 1+ x2 is given. We prove, for instance: π(3+8 √ 2)x 7+6 √ 1+ x2 +16 √ 2 √ 1+ x2 + √ 1+ x2 arctan x 45x 7+6 √ 1+ x2 +16 √ 2 √ 1+ x2 + √ 1+ x2 . Other algebraic approximations for the arctangent functions are, rather informally, presented. Mathematics subject classification (2010): 26D05.