Summary: In this study, we establish above and below bounds for the left and right hand sides of fractional Hermite-Hadamard inequalities by using functions whose second derivatives are bounded. We also give some refinements of fractional Hermite-Hadamard inequalities by using the functions that have the conditions \(f^{\prime}(a+b-t)-f^{\prime}(t)\geq 0,\) \(t\in \left[ a,\frac{a+b}{2}\right]\).