In this paper we present variety of Hardy-type inequalities and their refinements for an extension of Riemann-Liouville fractional derivative operators. Moreover, we use an extension of extended Riemann-Liouville fractional derivative and modified extension of Riemann-Liouville fractional derivative using convex and monotone convex functions. Furthermore, mean value theorems and n-exponential convexity of the related functionals is discussed.
G = G(x, y) = ?xy, L = L(x,y) = x?y/log(x)?log(y)' I=I(x,y)= 1/e(xx?yy)
1/(x-y) be the geometric, logarithmic, identric, and arithmetic means of x
and y. We prove that the inequalities L(G2,A2) < G(L2,I2) < A(L2,I2)
< I(G2,A2) are valid for all x, y > 0 with x ? y. This refines a result
of Seiffert.
The goal of the paper is to present some results concerning the approximation of convex functions by linear positive operators. First, one recalls some results concerning the univariate real valued convex functions. Next, one presents the notion of higher order convexity introduced by Popoviciu [Popoviciu, T., Sur quelques propri´et´ees des fonctions d’une ou deux variable r´eelles, PhD Thesis, La Faculte des Sciences de Paris, 1933
(June)] . The Popoviciu’s famous theorem for the representation of linear functionals associated to convex functions of m−th order (with the proof of author) is also presented. Finally, applications of the convexity to study the monotonicity of sequences of some linear positive operators and also mean value theorems for the remainder term of some approximation formulas based on linear positive operators are presented.
Local and global regularity properties of weak solutions of the Schrödinger equation −Δu+qu=λu play an important role in the spectral theory of the corresponding operator [Formula: see text]. Central among these properties is local boundedness of the solutions u, which is derived in an elementary way for potentials q whose negative parts q− lie in the local Kato class K loc . The method also provides mean value inequalities for and, in case q+ is in K loc too, continuity of u. To employ these mean value inequalities for bounds on eigenfunctions of T in a fixed direction, classes Kρ are introduced which reflect the behavior of q at infinity. A couple of examples allow to compare these classes with more conservative ones like the Stummel class Q and the global Kato class K. The fundamental property of local boundedness of solutions also serves as a base for a very short proof of the self-adjointness of T if the operator is bounded from below and q−∈K loc . If q(x) is permitted to go to −∞, as |x|→∞, a large class K ρ which guarantees self-adjointness of T is derived and contains the case q−(x)= O (|x|2). The Spectral Theorem then allows to deduce rapidly decaying bounds on eigenfunctions for discrete eigenvalues, at least if q−(x)= o (|x|2). This is also the condition under which the existence of a bounded solution is sufficient to guarantee λ∈σ(T). Here q−(x)= O (|x|2) appears as a borderline case and is discussed at some length by means of an explicit example. The class of admissible operators extending to these borderline cases with potentials singular locally and at infinity, the regularity results for solutions being mostly optimal, as demonstrated by numerous examples, yet the proofs being shorter and more straightforward than those to be found in literature for smaller classes and weaker results, the sets Kρ under consideration and the methods employed appear to be quite natural.
The aim of this paper is to establish some refined versions of majorization inequality involving twice differentiable convex functions by using Taylor theorem with mean-value form of the remainder. Our results improve several results obtained in earlier literatures. As an application, the result is used for deriving a new fractional inequality.
Characterization of Inner Product Spaces by Strongly Schur-Convex Functions
