Monotonicity rules for the ratio of two function series and two integral transforms

In this paper, we investigate the monotonicity of the functions t ↦ ∑ k = 0 ∞ a k w k ( t ) ∑ k = 0 ∞ b k w k ( t ) t \mapsto \frac {\sum _{k=0}^\infty a_k w_k(t)}{\sum _{k=0}^\infty b_k w_k(t)} and x ↦ ∫ α β f ( t ) w ( t , x ) d t ∫ α β g ( t ) w ( t , x ) d t x \mapsto \frac {\int _\alpha ^\beta f(t) w(t,x) \mathrm {d} t}{\int _\alpha ^\beta g(t) w(t,x) \mathrm {d} t} , focusing on case where the monotonicity of a k / b k a_k/b_k and f ( t ) / g ( t ) f(t)/g(t) change once. The results presented also provide insights into the monotonicity of the ratios of two power series, two Z \mathcal {Z} -transforms, two discrete Laplace transforms, two discrete Mellin transforms, two Laplace transforms, and two Mellin transforms. Finally, we employ these monotonicity rules to present several applications in the realm of special functions and stochastic orders.