In this paper, by constructing a new non-homogeneous kernel of mixed hyperbolic functions, we establish a new Hilbert-type integral inequality with the best constant factor. We also consider the equivalent form of the obtained inequality. Moreover, by using the rational fraction expansion of cotangent function and cosecant function, some special Hilbert's type inequalities with the constant factors related to the higher derivatives of cotangent function and cosecant function are presented.
Abstract. In this paper, a theorem related to the Hilbert-type inequality is corrected. By introducing parameters, and using Euler-Maclaurin summation formula, we give a discrete form of the Hilbert-type inequality involving a non-homogeneous kernel. Furthermore, we prove that our result is a concise generalization of the corrected theorem and some known results. As applications, some particular new results are presented.Mathematics subject classification (2010): 26D15.
By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.
In this paper, by the introduction of several parameters, we construct a new kernel function which is defined in the whole plane and includes some classical kernel functions. Estimating the weight functions with the techniques of real analysis, we establish a new Hilbert-type inequality in the whole plane, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of the partial fraction expansion of the tangent function, some special and interesting inequalities are presented at the end of the paper.
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