406 research outputs found
Unconditional and Conditional Large Gaps between the zeros of the Riemann Zeta-Function
In this paper, first by employing inequalities derived from the Opial
inequality due to David Boyd with best constant, we will establish new
unconditional lower bounds for the gaps between the zeros of the Riemann zeta
function. Second, on the hypothesis that the moments of the Hardy Z-function
and its derivatives are correctly predicted, we establish some explicit
formulae for the lower bounds of the gaps between the zeros and use them to
establish some new conditional bounds. In particular it is proved that the
consecutive nontrivial zeros often differ by at least 6.1392 (conditionally)
times the average spacing. This value improves the value 4.71474396 that has
been derived in the literature
Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales
We will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of Hölder's inequality, a simple consequence of Keller's chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities will be derived from our results as special cases
Oscillation of Second Order Nonlinear Dynamic Equations on Time Scales
By means of Riccati transformation techniques, we establish some oscillation criteria for a second order nonlinear dynamic equation on time scales in terms of the coefficients. We give examples of dynamic equations to which previously known oscillation criteria are not applicable
Sneak-Out Principle on Time Scales
In this paper, we show that the so-called sneak-out principle for discrete inequalities is valid also on a general time scale. In particular, we prove some new dynamic inequalities on time scales which as special cases contain discrete inequalities obtained by Bennett and Grosse-Erdmann. The main results also are used to formulate the corresponding continuous integral inequalities, and these are essentially new. The techniques employed in this paper are elementary and rely mainly on the time scales integration by parts rule, the time scales chain rule, the time scales Hölder inequality, and the time scales Minkowski inequality
Gehring Inequalities on Time Scales
In this paper, we first prove a new dynamic inequality based on an application of the time scales version of a Hardy-type inequality. Second, by employing the obtained inequality, we prove several Gehring-type inequalities on time scales. As an application of our Gehring-type inequalities, we present some interpolation and higher integrability theorems on time scales. The results as special cases, when the time scale is equal to the set of all real numbers, contain some known results, and when the time scale is equal to the set of all integers, the results are essentially new
Kamenev type oscillation criteria for nonlinear difference equations
summary:By means of Riccati transformation techniques, we establish some new oscillation criteria for second-order nonlinear difference equation which are sharp
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