Mean Value Inequalities

1977 ◽  
Vol 8 (5) ◽  
pp. 785-791 ◽  
Author(s):  
A. M. Fink
Keyword(s):  
Mean Value ◽  
Mean Value Inequalities

scholarly journals A chain of mean value inequalities

2020 ◽  
pp. 27-27
Author(s):  
Horst Alzer ◽  
Man Kwong
Keyword(s):  
Mean Value ◽  
Arithmetic Means ◽  
A Chain ◽  
Mean Value Inequalities

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.

REGULARITY OF SOLUTIONS FOR SINGULAR SCHRÖDINGER EQUATIONS

1992 ◽  
Vol 04 (01) ◽  
pp. 95-161 ◽  
Author(s):  
ANDREAS M. HINZ
Keyword(s):  
Global Regularity ◽  
Fundamental Property ◽  
Mean Value ◽  
Borderline Case ◽  
Spectral Theorem ◽  
Regularity Of Solutions ◽  
Local Boundedness ◽  
Kato Class ◽  
Regularity Properties ◽  
Mean Value Inequalities

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.

scholarly journals Mean value inequalities and conditions to extend Ricci flow

2015 ◽  
Vol 22 (2) ◽  
pp. 417-438 ◽  
Author(s):  
Xiaodong Cao ◽  
Hung Tran
Keyword(s):  
Ricci Flow ◽  
Mean Value ◽  
Mean Value Inequalities

Nonsmooth Constrained Optimization and Multidirectional Mean Value Inequalities

1999 ◽  
Vol 9 (3) ◽  
pp. 690-706 ◽  
Author(s):  
Didier Aussel ◽  
Jean-Noël Corvellec ◽  
Marc Lassonde
Keyword(s):  
Constrained Optimization ◽  
Mean Value ◽  
Mean Value Inequalities

scholarly journals MEAN VALUE TYPE INEQUALITIES FOR QUASINEARLY SUBHARMONIC FUNCTIONS

2013 ◽  
Vol 55 (2) ◽  
pp. 349-368 ◽  
Author(s):  
OLEKSIY DOVGOSHEY ◽  
JUHANI RIIHENTAUS
Keyword(s):  
Sufficient Conditions ◽  
Continuous Functions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Subharmonic Functions ◽  
Mean Value Inequality ◽  
The Mean ◽  
Mean Value Inequalities ◽  
Necessary And Sufficient ◽  
Bounded Sets

AbstractThe mean value inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalise subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the characterisation of non-negative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalised mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the radius of this ball.

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