On deviations and strong asymptotic functions of meromorphic functions of finite lower order

Let f be a transcendental meromorphic function of finite lower order with N ( r , f ) = S ( r , f ) , and let q ν be distinct rational functions, 1 ⩽ ν ⩽ k . For 0 < γ < ∞ put B ( γ ) : = { π γ sin π γ if γ ⩽ 0.5 , π γ if γ > 0.5 . The estimate of the lower and upper logarithmic density of the set E ( γ ) = { r : ∑ 1 ⩽ ν ⩽ k log + max | z | = r | f ( z ) − q ν ( z ) | − 1 < B ( γ ) T ( r , f ) } is presented in the paper. For a transcendental meromorphic function f of finite lower order, the estimate of the lower and upper logarithmic density of the set E ˆ ( γ ) = { r : ∑ 1 ⩽ ν ⩽ k log + max | z | = r | f ( z ) − p ν ( z ) | − 1 < ( d + 2 ) B ( γ ) T ( r , f ) } is also presented, where p ν are distinct polynomials and d : = max 1 ⩽ ν ⩽ k deg ( p ν ) . Moreover, the notions of strong asymptotic rational function and strong rational asymptotic spot are defined. The results for E ( γ ) and E ˆ ( γ ) are applied to obtain upper estimates of the number of strong asymptotic rational functions and strong rational asymptotic spots of meromorphic functions.