In this work, a weighted H lder function that approximates a Jacobi polynomial which solves the second order singular Sturm-Liouville equation is discussed. This is generally equivalent to the Jacobean translations and the moduli of smoothness. This paper aims to focus on improving methods of approximation and finding the upper and lower estimates for the degree of approximation in weighted H lder spaces by modifying the modulus of continuity and smoothness. Moreover, some properties for the moduli of smoothness with direct and inverse results are considered.

Free vibration behavior was developed under the ratio of critical buckling temperature of laminated composite thin plates with the general elastic boundary condition. The equations of motion were found based on classical laminated plate theory (CLPT) while the solution functions consists of trigonometric function and a continuous function that is added to guarantee the sufficient smoother of the so-named remaining displacement function at the boundaries, in this research, a modified Fourier series were used, a generalized procedure solution was developed using Ritz method combined with the imaginary spring technique. The influences of many design parameters such as angles of layers, aspect ratio, thickness ratio, and ratio of initial in-