In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities, we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson’s inequalities, and quantum Newton’s inequalities. We also give some new midpoint-type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.
Application of Möbius coordinate transformation in evaluating Newton's integralWe propose a numerical scheme which efficiently combines various existing methods of solving the Newton's volume integral. It utilises the analytical solution of Newton's integral for tesseroid in computing the near-zone contribution to gravitational field quantities (potential and its first radial derivative). The far-zone gravitational contribution is computed using the expressions derived based on applying Molodensky's truncation coefficients to a spectral representation of Newton's integral. The weak singularity of Newton's integral is treated analytically using formulas for the gravitational contribution of the cylindrical mass volume centered with respect to the observation point. All three solutions are defined and evaluated in the system of polar spherical coordinates. A conversion of the geographical to polar spherical coordinates of input data sets (digital terrain and density models) is based on the Möbius transformation with an enhanced integration grid resolution at vicinity of the observation point.
Some Parameterized Quantum Simpson’s and Quantum Newton’s Integral Inequalities via Quantum Differentiable Convex Mappings
