On deviations from rational functions of entire functions of finite lower order

Let $f$ be a transcendental entire function of finite lower order, and let $q_\nu$ be rational functions. For $0<\gamma<\infty\,$ let $$ B(\gamma):=\cases{ {\displaystyle\frac{\pi\gamma}{\sin\pi\gamma}}&\hbox{if }\gamma\leq 0.5,\cr \pi\gamma&\hbox{if }\ga