Effective exponential bounds on the prime gaps

Over the last 50 years a large number of effective exponential bounds on the first Chebyshev function $\vartheta (x)$ have been obtained. Specifically we shall be interested in effective exponential bounds of the form $$\mathrm{\mid }\vartheta (x)-x\mathrm{\mid }<ax(\ln x{)}^b\exp (-c\sqrt{\ln x});(x\ge x_0)\mathrm{.}$$ Herein we shall convert these effective bounds on $\vartheta (x)$ into effective exponential bounds on the prime gaps $g_n=p_{n+1}-p_n$. Specifically we shall establish a number of effective exponential bounds of the form $$\frac{g_n}{p_n}<\frac{2a(\ln p_n{)}^b\exp (-c\sqrt{\ln p_n})}{1-a(\ln p_n{)}^b\exp (-c\sqrt{\ln p_n})};(x\ge x_{\ast });$$ and $$\frac{g_n}{p_n}<3a(\ln p_n{)}^b\exp (-c\sqrt{\ln p_n});(x\ge x_{\ast });$$ for some effective computable $x_{\ast }$. It is the explicit presence of the exponential factor, with known coefficients and known range of validity for the bound, that makes these bounds particularly interesting.