Showing papers in "Mathematical Inequalities & Applications in 2007"
TL;DR: In this paper, two Wilker-type inequalities involving hyperbolic functions are established for Mathematics subject classification (2000): 26D05, 26D15, and 26D16.
Abstract: In this note, two Wilker-type inequalities involving hyperbolic functions are established. Mathematics subject classification (2000): 26D05, 26D15.
70 citations
TL;DR: In this paper, it was shown that a variable exponent Bessel potential space coincides with the variable exponent Sobolev space if the Hardy-Littlewood maximal operator is bounded on the underlying variable exponent Lebesgue space.
Abstract: We show that a variable exponent Bessel potential space coincides with the variable exponent Sobolev space if the Hardy-Littlewood maximal operator is bounded on the underlying variable exponent Lebesgue space. Moreover, we study the Holder type quasi-continuity of Bessel potentials of the first order. Mathematics subject classification (2000): 46E35, 46E30, 26D10. Keywordsandphrases: Besselpotential space,Lebesguespacewithvariable exponent, quasi-continuity.
56 citations
TL;DR: In this article, a sharp triangle inequality and its reverse inequality with n elements in a Banach space X were presented, where equality attainedness was characterized for given x1, x2,..., xn in X, where equality attainness was given.
Abstract: We shall present a sharp triangle inequality and its reverse inequality with n elements in aBanach space X , or equivalently we shall estimate the difference ∑n j=1 ‖xj‖−‖ ∑n j=1 xj‖ for given x1, x2, . . . , xn in X , where equality attainedness will be characterized. Several applications will be given. Mathematics subject classification (2000): 46B20, 46B99.
53 citations
TL;DR: In this article, a group of chains of inequalities for homogeneous means are established, which generalize, strengthen and unify Tong-po Ling and Stolarsky inequalities, and a reversed chain of inequalities is derived.
Abstract: Suppose f (x, y) is a positive homogeneous function defined on U( R+ × R+) , then call ( f (ap ,bp) f (aq ,bq) ) 1 p−q two-parameter homogeneous function and denote by Hf (a, b; p, q) . If f (x, y) is third differentiable, then the log-convexity with respect to parameters p and q of Hf (p, q) depend on the sign of J = (x− y)(xI)x , where I = (ln f )xy . As applications a group of chains of inequalities for homogeneous means are established, which generalize, strengthen and unify Tong-po Ling ’s and Stolarsky’s inequalities, and a reversed chain of inequalities for exponential mean (identic mean) is derived, which contains a reversed Stolarsky’s inequality. Several estimations of lower and upper bounds of extended mean are presented. Mathematics subject classification (2000): 26B25, 26D07, 26E60,26A48.
36 citations
TL;DR: In this article, the logarithmic mean is proved to be completely monotonically monotonic and an open problem about the complete monotonicity of the extended mean values is posed.
Abstract: In the article, the logarithmic mean is proved to be completely
monotonic and an open problem about the logarithmically complete monotonicity
of the extended mean values is posed.
34 citations
TL;DR: In this paper, a generalization of the Dunkl-Williams inequality for finitely many elements in a normed linear space is presented. But this generalization does not consider the case of elements of a strictly convex normed space.
Abstract: In this paper we establish a generalization of the Dunkl-Williams inequality for finitely many elements in a normed linear space. As a consequence, we get some recently obtained results on the generalized triangle inequality and its reverse inequality. The case of equality for elements of a strictly convex normed linear space is also considered. Mathematics subject classification (2000): 26D15.
32 citations
31 citations
TL;DR: In this article, it was shown that under some conditions imposed on p, b, and c, the Askey inequality holds true for the normalized generalized Bessel function λp, i.e., that λ p(x) + εp(y) ≤ 1 +λp(z), where x, y ≥ 0 and z = x + y.
Abstract: Let up denote the normalized, generalized Bessel function of order p which depends on two parameters b and c and let λp(x) = up(x), x ≥ 0. It is proven that under some conditions imposed on p, b, and c the Askey inequality holds true for the function λp , i.e., that λp(x) +λp(y) ≤ 1 +λp(z), where x, y ≥ 0 and z = x + y. The lower and upper bounds for the function λp are also established.
31 citations
TL;DR: In this article, a simple norm inequality is proposed for inner product spaces. But it does not describe the inner product space of linear differential equations and functional analysis, and it is difficult to prove its correctness.
Abstract: [1] C. F. DUNKL, K. S. WILLIAMS, A simple norm inequality, Amer. Math. Monthly, 71, (1964), 53–54. [2] L. MALIGRANDA, Simple norm inequalities, Amer. Math. Monthly, 113, (2006), 256–260. [3] J. L. MASSERA, J. J. SCHAFFER, Linear differential equations and functional analysis I, Ann. of Math., 67, (1958), 517–573. [4] W. A. KIRK, M. F. SMILEY, Another characterization of inner product spaces, Amer. Math. Monthly, 71, (1964), 890–891.
28 citations
TL;DR: In this article, a new symmetric function, which generalizes Hamy symmetric functions, is defined and its properties, including Schur-geometric convexity, are investigated.
Abstract: A new symmetric function, which generalizes Hamy symmetric function, is defined. Its properties, including Schur-geometric convexity, are investigated. Some analytic inequalities are also established. Mathematics subject classification (2000): 26A51, 26D15, 0E05.
28 citations
TL;DR: In this paper, the authors consider the weighted Hardy inequality 1/q 1/p q ∞ ∞ x f p (x)v(x)dx f (t)dt u(x)-dx C 0 0 0 1 for the case 0 1.
Abstract: We consider the weighted Hardy inequality 1/q 1/p q ∞ ∞ x f p (x)v(x)dx f (t)dt u(x)dx C 0 0 0for the case 0 1 . The weights u(x) and v(x) for which this inequalityholds for all f (x) 0 may be characterized by the Mazya-Rosin or by the Persson-Stepanovconditions. In this paper, we show that these conditions are not unique and can be supplementedby some continuous scales of conditions and we prove their equivalence. The results for the dualoperator which do not follow by duality when 0 < q < 1 are also given.
TL;DR: In this paper, the authors introduce three different proofs of Priceos inequality and some new trigonometric and hyperbolic inequalities, and introduce a trigonometrical version of the Priceos inequalities.
Abstract: We introduce three different proofs of Priceos inequality and some new trigonometric and hyperbolic inequalities.
TL;DR: In this article, a theorem on |N̄, pn|k summability factors of infinite series has been proved under more weaker conditions, and a new result concerning the |C, 1|k sumability factors has been obtained.
Abstract: In the present paper a theorem on |N̄, pn|k summability factors of infinite series has been proved under more weaker conditions. Also we have obtained a new result concerning the |C, 1|k summability factors. Mathematics subject classification (2000): 40D15, 40F05, 40G99.
TL;DR: In this paper, the generalized (extended) logarithmic mean of two positive numbers (a and b) is defined as a function f(r) = Lr(a,b) Lr (a,c) is strictly decreasing on (−∞,∞), where L r denotes the generalized logarity of two numbers a and b.
Abstract: Let c > b > a > 0 be real numbers. Then the function f (r) = Lr(a,b) Lr(a,c) is strictly decreasing on (−∞,∞) , where Lr(a, b) denotes the generalized (extended) logarithmic mean of two positive numbers a and b . Mathematics subject classification (2000): 26D15, 26E60.
TL;DR: In this article, it was shown that all zeros of the reciprocal polynomial can be shown to be the same as all zero numbers of the corresponding non-negative polynomials.
Abstract: The purpose of this paper is to show that all zeros of the reciprocal polynomial
TL;DR: In this article, the Shafer, Fink and Malesevic inequalities are deduced from the solution of Oppeheim's problem, and the open problem is discussed carefully.
Abstract: In this paper, the open problem proposed by Oppeheim [A. Oppeheim, E1277, The American Mathematical Monthly, 64, (1957) p. 504] is discussed carefully. At the same time, the Shafer, Fink and Malesevic inequalities are deduced from the solution of Oppeheim’s problem. Mathematics subject classification (2000): 26D15.
TL;DR: In this paper, the λ-method of Mitrinovic-Vasic is applied to improve the upper bound for the arc sin function of L. Zhu, which belongs to R. E. Shafer.
Abstract: In this article λ-method of Mitrinovic-Vasic [1] is applied to improve the upper bound for the arc sin function of L. Zhu [4]. 1. Inequalities of Shafer-Fink’s type D. S. Mitrinovic in [1] considered the lower bound of the arc sin function, which belongs to R. E. Shafer. Namely, the following statement is true. Theorem 1.1 For 0 ≤ x ≤ 1 the following inequalities are true: 3x 2 + √ 1− x2 ≤ 6( √ 1 + x− √ 1− x) 4 + √ 1 + x+ √ 1− x ≤ arc sinx . (1) A.M. Fink proved the following statement in [2] . Theorem 1.2 For 0 ≤ x ≤ 1 the following inequalities are true: 3x 2 + √ 1− x2 ≤ arc sinx ≤ πx 2 + √ 1− x2 . (2) B. J. Malesevic proved the following statement in [3]. Theorem 1.3 For 0 ≤ x ≤ 1 the following inequalities are true: 3x 2 + √ 1− x2 ≤ arc sinx ≤ π π − 2 x 2 π − 2 + √ 1− x2 ≤ πx 2 + √ 1− x2 . (3) The main result of the article [3] can be formulated with the next statement. Proposition 1.4 In the family of the functions: fb(x) = (b+ 1)x b + √ 1− x2 (0 ≤ x ≤ 1), (4) according to the parameter b > 0, the function f2(x) is the greatest lower bound of the arc sin x function and the function f2/(π−2)(x) is the least upper bound of the arc sin x function. Research partially supported by the MNTRS, Serbia, Grant No. 144020.
TL;DR: In this paper, the authors studied the relationship between smoothness properties of a 2π -periodic function and the asymptotic behavior of its Fourier coefficients and expressed the smoothness conditions of a function in terms of it being in the generalized Lipschitz classes.
Abstract: We study the relationships between smoothness properties of a 2π -periodic function and the asymptotic behavior of its Fourier coefficients. The smoothness conditions of a function are expressed in terms of it being in the generalized Lipschitz classes. Mathematics subject classification (2000): 26A16, 42A16.
TL;DR: In this paper, a Dirichlet problem relative to the equation Lu = gφ− (f iφ)xi, where L is a linear elliptic operator with lower-order terms whose ellipticity condition is given in terms of the function φ(x) = (2π)− 2 exp ( − |x| /2 ), the density of the Gaussian measure.
Abstract: In this paper we study a Dirichlet problem relative to the equation Lu = gφ− (f iφ)xi , where L is a linear elliptic operator with lower-order terms whose ellipticity condition is given in terms of the function φ(x) = (2π)− 2 exp ( − |x| /2 ) , the density of the Gaussian measure. We use the notion of rearrangement with respect to the Gauss measure to obtain a priory estimate of the solution u and we study the summability of u in the Lorentz-Zygmund spaces when g and f i varies in suitable Lorent-Zygmund spaces. Mathematics subject classification (2000): 35J70, 35B65.
TL;DR: This chapter establishes a series of fuzzy random Shisha–Mond type inequalities of L q -type 1 ≤ q < ∞ and related fuzzy random Korovkin type theorems, regarding the fuzzy random q-mean convergence of fuzzyrandom positive linear operators to the fuzzyrandom unit operator for various cases.
Abstract: Here we study the fuzzy random positive linear operators acting on fuzzy random continuous functions. We establish a series of fuzzy random Shisha–Mond type inequalities of L q -type 1 ≤ q < ∞ and related fuzzy random Korovkin type theorems, regarding the fuzzy random q-mean convergence of fuzzy random positive linear operators to the fuzzy random unit operator for various cases. All convergences are with rates and are given using the above fuzzy random inequalities involving the fuzzy random modulus of continuity of the engaged fuzzy random function. The assumptions for the Korovkin theorems are minimal and of natural realization, fulfilled by almost all example fuzzy random positive linear operators. The astonishing fact is that the real Korovkin test functions assumptions are enough for the conclusions of the fuzzy random Korovkin theory. We give at the end applications. This chapter follows [22].
TL;DR: In this paper, a new class of generalized nonlinear multi-valued quasi-variational-like inclusions with H-monotone operators in Hilbert spaces is introduced and studied.
Abstract: In this paper, we introduce and study anew class of generalized nonlinear multi-valued quasi-variational-like inclusions With H-monotone operators in Hilbert spaces. By using the resolvent operator method associated with H-monotone operator due to Fang and Huang, we construct a new iterative algorithm for solving this kind of nonlinear multi-valued variational inclusions. We also prove the existence of solutions for the nonlinear multi-valued variational inclusions and the convergence of iterative sequences generated by the algorithm. Our results improve and generalize many known corresponding results.
TL;DR: In this paper, the necessary and sufficient condition for the embedding of discrete weighted Lebesgue spaces with variable exponents is given for subject classification in Mathematics Subject Classification (MPC).
Abstract: Given mappings p, q, v, w : Z → (0,∞) we can consider discrete weighted Lebesgue spaces {pn}(vn) and {qn}(wn) with variable exponents. The necessary and sufficient condition to the p , q , v , w for the embedding {pn}(vn) ↪→ {qn}(wn) is given. Mathematics subject classification (2000): 46E30, 26D15.
TL;DR: Upper bounds for the probability of the union of events based on the individual probabilities and joint probabilities of pairs are presented and can be interpreted as objective function values corresponding to feasible solutions of the dual of the Boolean probability bounding LP.
Abstract: We present upper bounds for the probability of the union of events based on the individual probabilities and joint probabilities of pairs. The bounds generalize Hunter’s upper bound and can be interpreted as objective function values corresponding to feasible solutions of the dual of the Boolean probability bounding LP. Mathematics subject classification (2000): 90C35, 90C90, 90C05.
TL;DR: In this paper, a generalization of the Furuta inequality is proposed, where A,B are J -selfadjoint matrices with nonnegative eigenvalues and I JA JB.
Abstract: In this article, we study matrix inequalities on an (indefinite) inner product space, including a generalization of Furuta inequality: let A,B be J -selfadjoint matrices with nonnegative eigenvalues and I JA JB. Then for each r 0 , (A r 2 ApA r 2 ) 1 q J(A r 2 BpA 2 ) 1 q holds for p 0, q 1 with (1 + r)q p + r. Mathematics subject classification (2000): 47B50, 47A63.
TL;DR: In this paper, a generalized mixed set-valued variational inequality problem is considered and a two-step iterative algorithm and an inertial proximal iterative method are proposed.
Abstract: In this paper, we consider a generalizedmixed set-valued variational inequality problem which includes many important known variational inequality problems and equilibrium problem, and its related some auxiliary variational inequality problems. We prove the existence of solutions of the auxiliary variational inequality problems and suggest a two-step iterative algorithm and an inertial proximal iterative algorithm. Further, we discuss the convergence analysis of iterative algorithms. The theorems presented in this paper improve and generalize many known results for solving equilibrium problems, variational inequality and complementarity problems in the literature. Mathematics subject classification (2000): 47H04, 47H06, 47H10, 47J20, 47J25, 47J30, 49J40.
TL;DR: For orthoprojectors of rank one, this article obtained a Chebyshev type inequality and a Gruss-Lupas type inequality for rank one ortho-projectors.
Abstract: In this paper we prove an inequality for certain orthoprojectors. For orthoprojectors of rank one we obtain a Chebyshev type inequality. Gruss-Lupas type inequalities are also discussed. Mathematics subject classification (2000): 26D15, 26D20, 15A39, 06F20.
TL;DR: In this article, a Durrmeyer type integral modification of the wellknown Baskakov operators with the weight function of the Beta basis function was studied, and the authors established an asymptotic formula and error estimation in terms of higher order modulus of continuity.
Abstract: In the present paper, we study a Durrmeyer type integral modification of the wellknown Baskakov operators with the weight function of Beta basis function. Some approximation properties of these operators were recently studied by Finta [2]. Here we study simultaneous approximation properties for these operators. We estimate local direct result in terms of modulus of continuity. The operators considered in this paper reproduce not only the constant functions but also the linear ones, due to this property we can improve the order of approximation for these operators by applying the iterative combinations, which were first studied by Micchelli [7]. We establish an asymptotic formula and error estimation in terms of higher order modulus of continuity in simultaneous approximation for the Micchelli combinations of these operators. Mathematics subject classification (2000): 41A30, 41A36.
TL;DR: In this article, a non-negative triangular matrix operator is considered in weighted Lebesgue spaces of sequences and some new weight characterizations for discrete Hardy type inequalities with matrix operator are proved for the case 1 < q < p < ∞.
Abstract: A non-negative triangular matrix operator is considered in weighted Lebesgue spaces ofsequences. Under some additional conditions on the matrix, some new weight characterizationsfor discrete Hardy type inequalities with matrix operator are proved for the case 1 < q < p < ∞.Some further results are pointed out.
TL;DR: In this article, it was shown that functions in Hα (β) are univalent for all real numbers α and β satisfying α β < 1 and that the result is sharp in the sense that the constant β cannot be replaced by any real number less than α.
Abstract: Let Hα (β) denote the class of normalized functions f , analytic in the unit disc E , which satisfy the condition Re [ (1 − α)f ′(z) + α ( 1 + zf ′′(z) f ′(z) )] > β , z ∈ E, where α and β are pre-assigned real numbers. H. S. Al-Amiri and M. O. Reade, in 1975, have shown that for α 0 and also for α = 1 , the functions in Hα (0) are univalent in E . In 2005, V. Singh, S. Singh and S. Gupta proved that for 0 < α < 1 , functions in Hα (α) are also univalent. In the present note, we establish that functions in Hα (β) are univalent for all real numbers α and β satisfying α β < 1 and that the result is sharp in the sense that the constant β cannot be replaced by any real number less than α . Mathematics subject classification (2000): 30C45, 30C50.
TL;DR: The problem of the Hyers-Ulam stability of the Hermite-Hadamard inequality posed by Zs. Páles is solved in this paper, and it is shown that if f is continuous and satisfies both of the above inequalities simultaneously, then it is 4 -convex.
Abstract: The problem of the Hyers-Ulam stability of the Hermite-Hadamard inequality posed by Zs. Páles is solved. It is shown that for continuous functions f : I → R neither the inequality f ( x+y 2 ) 1 y−x ∫ y x f (t) dt + nor 1 y−x ∫ y x f (t) dt f (x)+f (y) 2 + implies the c − convexity of f (with any c > 0 ). However, if f is continuous and satisfies both of the above inequalities simultaneously, then it is 4 -convex. Mathematics subject classification (2000): 39B82, 26A51.