In 1974, Hardy Rogers gave the following fixed point theorem,
Let $(M,d)$ be a complete metric space and $T$ a self-mapping of $M$ satisfying the following condition for $ x,y \in M, $
$$d(Tx,Ty) \leq ad (x,Tx) + bd(y,Ty) + cd (x,Ty) + ed(y,Tx) + fd(x,y)$$
where $a,b,c,e,f$ are non negative real numbers with $a+b+c+e+f<1$ then $T$ has a unique fixed point.
In his proof, he put $$ \beta = \min \Big(\frac{a+c+f} {1-b-c},\frac{b+e+f}{1-a-e}\Big) $$ and said that $\beta<1.$
I want to prove it. also i am sharing my tried proof