Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus

Abstract

Sharp bounds for cosh(x)x,sinh(x)x, and sin(x)x were obtained, as well as one new bound for ex+arctan(x)x. A new situation to note about the obtained boundaries is the symmetry in the upper and lower boundary, where the upper boundary differs by a constant from the lower boundary. New consequences of the inequalities were obtained in terms of the Riemann–Liovuille fractional integral and in terms of the standard integral.

Integral Inequalities Involving Strictly Monotone Functions

“…Functional inequalities involving special functions are very useful in mathematical analysis, and several interesting results have been obtained in this topic. See e.g., [2,[15][16][17][18][19][20][21][22][23][24][25].…”

On Some Important Class of Dynamic Hilbert’s-Type Inequalities on Time Scales

“…Over the past decade, a great number of dynamic Hilbert-type inequalities on time scales have been established by many researchers who were motivated by some applications; see the papers [2][3][4][5][6][7][8][9][10][11][12][13]. For more details on time scales calculus, see [14].…”