Ando–Hiai type inequalities for multivariate operator means

ABSTRACT

We present several Ando–Hiai type inequalities for n-variable operator means for positive invertible operators. Ando–Hiai's inequalities given here are not only of the original type but also of the complementary type and of the reverse type involving the generalized Kantorovich constant.     

Some Hermite–Hadamard type inequalities for functions of generalized convex derivative

We obtain some new inequalities of Hermite–Hadamard type. We consider functions that have convex or generalized convex derivative. Additional inequalities are proven for functions whose second derivative in absolute values are convex. Applications of the main results are presented.

     

Some Hermite–Hadamard type inequalities for geometrically quasi-convex functions

In the paper, we introduce a new concept ‘geometrically quasi-convex function’ and establish some Hermite–Hadamard type inequalities for functions whose derivatives are of geometric quasi-convexity.     

Koksma–Hlawka type inequalities of fractional order

The Koksma–Hlawka inequality states that the error of numerical integration by a quasi-Monte Carlo rule is bounded above by the variation of the function times the star-discrepancy. In practical applications though functions often do not have bounded variation. Hence here we relax the smoothness assumptions required in the Koksma–Hlawka inequality. We introduce Banach spaces of functions whose fractional derivative of order is in . We show that if α is an integer and p = 2 then one obtains the usual Sobolev space. Using these fractional Banach spaces we generalize the Koksma–Hlawka inequality to functions whose partial fractional derivatives are in . Hence we can also obtain an upper bound on the integration error even for certain functions which do not have bounded variation but satisfy weaker smoothness conditions.      

Some subordination results for classes of analytic functions defined by the Al-Oboudi–Al-Amoudi operator

In this paper, we investigate several interesting subordination results of some classes of analytic functions defined by means of the Al-Oboudi–Al-Amoudi operator. Received: 15 August 2008, Revised: 19 November 2008     

Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces

Some Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications in relation with the celebrated Hölder–McCarthy’s inequality for positive operators and Ky Fan’s inequality for real numbers are given as well.     

Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities

We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa–Leindler and Brascamp–Lieb inequalities. Further issues that we transpose to this functional setting are integral-geometric formulae of Cauchy–Kubota type, valuation property and isoperimetric/Urysohn-like inequalities.     

q–differ-integral operator on p–valent functions associated with operator on Hilbert space

Making use of multivalent functions with negative coefficients of the type f (z)=z^p-^∑∞_k=p+1a_kz^k,which are analytic in the open unit disk and applying the q-derivative a q–differintegral operator is considered.Furthermore by using the familiar Riesz-Dunford integral,a linear operator on Hilbert space H is introduced.A new subclass of p-valent functions related to an operator on H is defined.Coefficient estimate,distortion bound and extreme po...     

Turán type inequalities for Krätzel functions

Complete monotonicity, Laguerre and Turán type inequalities are established for the so-called Krätzel function $Z_{\rho }^{\nu }$, defined by$Z_{\rho }^{\nu }(u)=\underset{0}{\overset{\infty }{\int }}{t}^{\nu ?1}{e}^{?t^{\rho }?\frac{u}{t}}\mathrm{d}t,$ where $u>0$ and $\rho ,\nu \in \mathbb{R}$. Moreover, we prove the complete monotonicity of a determinant function of which entries involve the Krätzel function.     

Some properties of certain multivalent analytic functions involving the Cho–Kwon–Srivastava operator

By making use of the principle of subordination between analytic functions and the Cho–Kwon–Srivastava operator, we introduce a certain subclass of multivalent analytic functions. Such results as subordination and superordination properties, convolution properties, inclusion relationships, distortion theorems, inequality properties and sufficient conditions for multivalent starlikeness are proved. The results presented here would provide extensions of those given in earlier works. Several other new results are also obtained.     

Čebyšev's type inequalities for functions of selfadjoint operators in Hilbert spaces

Some inequalities for continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given.     

Hilbert–Pachpatte type fractional integral inequalities

We present here very general weighted univariate and multivariate Hilbert–Pachpatte type integral inequalities. These involve Caputo and Riemann–Liouville fractional derivatives and fractional partial derivatives of the mentioned types.     

Unitarily invariant norm inequalities for accretive–dissipative operator matrices

An open question raised in Lin and Zhou (2013) [13] is answered.     

Hilbert–Pachpatte type general integral inequalities

In this article we present very general weighted Hilbert–Pachpatte type integral inequalities. These are regarding ordinary derivatives and fractional derivatives of Riemann–Liouiville and Canavati types. Also regarding general derivatives of Widder type and linear differential operators. Our results apply to continuous functions and some to integrable functions.