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For nonzero vectors x and y in the normed linear space (X, ‖ ⋅ ‖), we can define the p-angular distance by $$ {\alpha}_p\left[x,y\right]:=\left\Vert {\left\Vert x\right\Vert}^{p-1}x-{\left\Vert y\right\Vert}^{p-1}y\right\Vert . $$ We show (among other results) that, for p ≥ 2, $$ \begin{array}{l}{\alpha}_p\left[x,y\right]\le p\left\Vert y-x\right\Vert {\displaystyle \underset{0}{\overset{1}{\int }}{\left\Vert \left(1-t\right)x+ty\right\Vert}^{p-1}dt}\hfill \\ {}\kern3.36em \le p\left\Vert y-x\right\Vert \left[\frac{{\left\Vert x\right\Vert}^{p-1}+{\left\Vert y\right\Vert}^{p-1}}{2}+{\left\Vert \frac{x+y}{2}\right\Vert}^{p-1}\right]\hfill \\ {}\kern3.36em \le p\left\Vert y-x\right\Vert \frac{{\left\Vert x\right\Vert}^{p-1}+{\left\Vert y\right\Vert}^{p-1}}{2}\le p\left\Vert y-x\right\Vert {\left[ \max \left\{\left\Vert x\right\Vert, \left\Vert y\right\Vert \right\}\right]}^{p-1},\hfill \end{array} $$ for any x, y ∈ X. This improves a result of Maligranda from [“Simple norm inequalities,” Amer. Math. Month., 113, 256–260 (2006)] who proved the inequality between the first and last terms in the estimation presented above. The applications to functions f defined by power series in estimating a more general “distance” ‖f(‖x‖)x − f(‖y‖)y‖ for some x, y ∈ X are also presented.

Refinements of the generalized trapezoid inequality for functions of bounded variation in terms of the cumulative variation function are given. Applications for selfadjoint operators on complex Hilbert spaces are also provided.

Some inequalities of Hermite-Hadamard type for λ-convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.

In the present paper, a coupled algorithm refining recursively the Hermite–Hadamard inequality on a simplex is investigated. Our approach allows us to express the integral mean value $M_{f}$ of a convex function $f$ on a simplex as both the limit of sequences and sum of series involving iterative lower and upper bounds of $M_{f}$. Two examples of interest are discussed.

We investigate the non-commutative -divergence functional of operator log-convex functions. In particular, we show that a continuous function is operator log-convex if and only iffor all continuous fields and of strictly positive operators. Moreover, we define operator -convex functions and investigate some of its properties. In particular, we show that this class contains operator convex functions and operator log-convex functions.

In this paper, new Hermite-Hadamard type inequalities for n-times differentiable log-preinvex functions are established. The established results generalize some of those results proved in recent papers for differentiable log-preinvex functions and differentiable log-convex functions.

Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral R b a f eit du (t) of continuous complex valued integrands f : C (0; 1)! C de…ned on the complex unit circle C (0; 1) and various subclasses of integrators u : [a; b] [0; 2 ] ! C of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well. 1. Introduction The problem of approximating the Riemann-Stieltjes integral R b a f (t) du (t) by the quantity f (x) [u (b) u (a)] ; which is a natural generalization of the Ostrowski problem analyzed in 1937 (see [17]), was apparently …rst considered in the literature by the author in 2000 (see [9]) where he obtained the following result: [u (b) u (a)] f (x) Z b a f (t) du (t) (1.1)

Some inequalities of Jensen type for h-convex functions defined on convex subsets in real or complex linear spaces are given. Applications for norm inequalities are provided as well.

This paper deals with some new theorems and inequalities about a Fejér type integral inequality which estimate the difference between the right and middle part in Fejér inequality with new bounds. Also some applications to higher moments of random variables, an error estimate for trapezoidal formula, and some inequalities in connection with special means are given.

We introduce the concept of operator h-convex functions for positive linear maps, and prove some Hermite-Hadamard type inequalities for these functions. As applications, we obtain several trace inequalities for operators.

A copula is a function which joins (or ‘couples’) a bivariate distribution function to its marginal (one-dimensional) distribution functions. In this paper, we obtain Chebyshev type inequalities by utilising copulas.

In this paper some generalizations of Buzano inequality for $n$-tuples of vectors in inner product spaces are given. Applications for norm and numerical radius inequalities for $n$-tuples of bounded linear operators and for functions of normal operators defined by power series with nonnegative coefficients are also provided.

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In this paper, we establish some companions of Fejer’s inequality for convex functions which generalize the inequalities of Hermite-Hadamard type from 'Two mappings in connection to Hadamard’s inequalities' (Dragomir, 1992) and 'On some refinements of Hadamard’s inequalities and applications' (Dragomir, Milosevic and J. Sandor,1993)

Companions of Hermite-Hadamard inequalities for convex functions defined on the positive axis in the case when the integral has either the weight ψ or 1 ,t > 0 are given. Applications for special means are provided as well.

In the present paper we investigate the problem of approximating the Riemann-Stieltjes integral ? a b f ( λ ) du ( λ ) in the case when the integrand f is ( n + 1 ) -time differentiable ( n ? 0 ) and the derivative f ( n + 1 ) is continuous on a , b , while the integrator u is Riemann integrable on a , b . A priory error bounds for different classes of functions are provided.

Some inequalities of Jensen type for Arg-square-convex functions of unitary operators in Hilbert spaces are given.

If $\{ E_{\lambda} \} _{\lambda\in\mathbb{R}}$ is the spectral family of a bounded selfadjoint operator A on a Hilbert space H and m=minSp(A) and M=maxSp(A), we show that for any continuous function φ: $[ m,M ] \rightarrow \mathbb{C}$ , we have the inequality $$\begin{aligned} \bigl\vert \bigl\langle \varphi ( A ) x,y \bigr\rangle \bigr\vert ^{2} \leq& \Biggl( \int_{m-0}^{M}\bigl\vert \varphi ( t ) \bigr\vert \,d \Biggl( \bigvee_{m-0}^{t} \bigl( \langle E_{ ( \cdot ) }x,y \rangle \bigr) \Biggr) \Biggr) ^{2} \\ \leq& \bigl\langle \bigl\vert \varphi ( A ) \bigr\vert x,x \bigr\rangle \bigl\langle \bigl\vert \varphi ( A ) \bigr\vert y,y \bigr\rangle \end{aligned}$$ for any vectors x and y from H. Some related results and applications are also given.

In this paper we establish some Lipschitz type inequalities for complex functions when some convexity properties for powers of the absolute value of the derivative are assumed. The case of functions defined by power series is analyzed. Some applications in bounding the Jensen difference for complex functions are also provided.

A new generalization of the Ostrowski–Gruss inequality is introduced in three different cases for functions in L1[a, b] and L∞[a, b] spaces and its application is given for deriving error bounds of some quadrature rules.

In this paper we establish some new upper and lower bounds for the difference between the weighted arithmetic and harmonic operator means under various assumptions for the positive invertible operators A, B. Some applications when A, B are bounded above and below by positive constants are given as well.

In this paper, we obtain new inequalities for relative operator entropy [Formula: see text] in terms of Tsallis’ relative entropy [Formula: see text] [Formula: see text] in the case of positive invertible operators [Formula: see text] [Formula: see text]. Further bounds for [Formula: see text] are also provided.

In this paper, a condition which implies the continuity of an [Formula: see text]-convex function is investigated. In fact, any [Formula: see text]-convex function bounded from above is continuous if the function [Formula: see text] satisfies a certain condition which is called pre-continuity condition.

For nonzero vectors x and y in the normed linear space (X,�·� ) we can define the p-angular distance by αp(x,y) := � � � xp−1 x −� yp−1 y � � . In this paper we show among others that 1 2

Some new inequalities of Ostowski-type for the Cebysev functional and applications for Taylor’s expansion and generalised trapezoid formula are pointed out. Mathematics subject classification (2010): Primary 26D15; Secondary 41A55.

Some new results related to the right-hand side of the Hermite- Hadamard type inequality for the class of functions whose derivatives at certain powers are s-convex functions in the second sense are obtained.

By the use of some integral inequalities containing superquadratic functions, we obtain an inequality which generalizes some previous results. We also present an inequality for positive linear mappings of operators on Hilbert spaces. Some applications and examples are given as well.

On utilizing a refinement and a reverse of Young's inequality, in this paper, we establish new inequalities for functions defined by power series with positive coefficients, which improve the famous Holder's inequality for power series. Some applications for special functions such as polylogarithm, hypergeometric and Bessel functions are presented as well.

For the @C"p","q-function, we derive several properties and characteristics related to convexity, log-convexity and complete monotonicity. Similar properties and characteristics of the corresponding (p,q)-analogue @j"p","q(x) of the digamma or the @j-function have also been established. By applying the main results in this paper when p->~ and q->1, we obtain all of the results given in several earlier works by (for example) Krasniqi, Shabani, and other authors. Some potential areas of applications of the results presented in this paper are also indicated.

Two new reverses of the celebrated Jensen integral inequality for convex functions with applications for means, the Holder inequality and f-divergence measures in information theory are given.

We obtain amongst others the following result for any x, y ∈ H, where is a function defined by power series with real coefficients and convergent on the open disk D(0, R) ⊂ ℂ, R > 0, , T ∈ ℬ(H), α, β ≥ 0 with α + β ≥ 1 and ‖T ‖2α, ‖T‖2β

In this paper, we derive new inequalities for power series with real coefficients and apply them for some complex functions of interest such as exponential, trigonometric, hypergeometric, polylogarithm and Bessel functions

Two new reverses of the celebrated Jensen’s inequality for convex functions in the general setting of the Lebesgue integral, with applications to means, Hölder’s inequality and$f$-divergence measures in information theory, are given.

A three point quadrature rule approximating the Riemann integral for a function of bounded variation f by a linear combination with real coefficients of the values f ( a ) , f ( x ) and f ( b ) with x ∈ [ a , b ] whose sum is equal to b − a is given. Applications for special means inequalities and in establishing a priori error bounds for the approximation of selfadjoint operators in Hilbert spaces by spectral families are provided as well.

In this paper we provide sharp error bounds in approximating the weighted Riemann–Stieltjes integral ∫ a b f ( t ) g ( t ) d α ( t ) by the weighted trapezoidal rule f ( a ) + f ( b ) 2 ∫ a b g ( t ) d α ( t ) . Applications for continuous functions of selfadjoint operators in complex Hilbert spaces are given as well.

Necessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

We extend some fundamental results about the Orlicz space LΦ (such as the Jensen and Holder inequalities, the LΦ Banach spaces and their duals) to Optimal Measure Theory.

Some inequalities of Gr¨ uss' type for vectors and continuous functions of selfadjoint operators in Hilbert spaces, under suitable assumptions for the involved operators, are given. Applications for power and logarithmic functions are provided as well.

© 2016 Vilnius Gediminas Technical University. Abstract: In this paper, we define some integral transforms and obtain suitable bounds for them in order to introduce a main class of integral inequalities including Ostrowski and Ostrowski-Grüss inequalities and various kinds of new integral inequalities. In this sense, we also introduce a three point quadrature formula and obtain its error bounds. ispartof: Mathematical Modelling and Analysis vol:21 issue:4 pages:569-584 status: published

In this paper we obtain some inequalities of Ostrowski and Hermite-Hadamard type for Lipschitzian mappings between two Banach spaces. Applications for functions of norms in Banach spaces and functions defined by power series in Banach algebras are provided as well.

Some new sharp Gruss’ type inequalities for functions of bounded variation and applications for selfadjoint operators in Hilbert spaces are given.

In this paper, we obtain some inequalities for functions defined by a power series with nonnegative coefficients. In order to obtain these inequalities, a refinement of the Schwarz inequality in inner product spaces is utilized. Natural applications for some elementary functions of interest are also provided.

In this paper, we establish some new refinements for the celebrated Fejer’s and Hermite-Hadamard’s integral inequalities for convex functions.