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On weak type inequalities for rare maximal functions

K. Hare, A. Stokolos (2000)

Colloquium Mathematicae

The properties of rare maximal functions (i.e. Hardy-Littlewood maximal functions associated with sparse families of intervals) are investigated. A simple criterion allows one to decide if a given rare maximal function satisfies a converse weak type inequality. The summability properties of rare maximal functions are also considered.

On Young's inequality.

Witkowski, Alfred (2006)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Opial inequalities on time scales

Martin Bohner, Bıllûr Kaymakçalan (2001)

Annales Polonici Mathematici

We present a version of Opial's inequality for time scales and point out some of its applications to so-called dynamic equations. Such dynamic equations contain both differential equations and difference equations as special cases. Various extensions of our inequality are offered as well.

Opial's type inequalities on time scales and some applications

S. H. Saker (2012)

Annales Polonici Mathematici

We prove some new Opial type inequalities on time scales and employ them to prove several results related to the spacing between consecutive zeros of a solution or between a zero of a solution and a zero of its derivative for second order dynamic equations on time scales. We also apply these inequalities to obtain a lower bound for the smallest eigenvalue of a Sturm-Liouville eigenvalue problem on time scales. The results contain as special cases some results obtained for second order differential...

Optimal sublinear inequalities involving geometric and power means

Jiajin Wen, Sui-Sun Cheng, Chaobang Gao (2009)

Mathematica Bohemica

There are many relations involving the geometric means G n ( x ) and power means [ A n ( x γ ) ] 1 / γ for positive n -vectors x . Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form ( 1 - λ ) G n γ ( x ) + λ A n γ ( x ) ≥ A n ( x γ ) and ( 1 - λ ) G n γ ( x ) + λ A n γ ( x ) ≤ A n ( x γ ) with parameters λ ∈ ℝ and γ ∈ ( 0 , 1 ) . We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are also demonstrated....

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