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Let G = (V , E) be a locally finite, connected and weighted graph. We prove that, for a graph satisfying curvature dimension condition C D E ′ (n , 0) and uniform polynomial volume growth of degree m, all non-negative solutions of the equation ∂ t u = Δ u + u 1 + α blow up in a finite time, provided that α = 2 m . We also consider the blow-up problem under certain conditions for volume growth and initial value. These results complement our previous work joined with Lin. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide new refinements of Becker-Stark inequality and Cusa-Huygens inequality using trigonometric polynomial method. It is shown that the approach proposed is useful for establishing new inequalities and refining some existing inequalities. [ABSTRACT FROM AUTHOR]