By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the rever...

For the data based choice of the bandwidth of a kernel density estimator, several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth. In the particular the relative rate of convergence is the square root of the sample size, which is known to be the possible. The point of this paper is to show how semiparametric arguments can be emp...

We study variants of the classic s-t cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC). • For any constant k ≥ 2 and ǫ > 0, we show that Directed Multicutwith k source-sink pairs is hard to approximate within a factor k − ǫ. This matches the trivial k-approximation algorithm. By a simple reduction, our result for k = 2 implies that Directed...

In this paper, we prove the Ulam stability of Fréchet functional equation f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(z+x) arising from characterization inner product spaces and determine its best constant. Using result, give a result for pexiderized version equation.