About Borel type relation for some positive functional series - Bodnarchuk, A.Yu.; Skaskiv, O.B.; Trusevych, O.M. (Matematychni Studii, 2025)
Let f be an entire transcendental function, (λ_n) be a non-decreasing to +∞ sequence, M_f(r)={|f(z)||z|=r}, and Γ_f(r)/r=(M_f(r))’_+ be a right derivative, r>0. For a regularly convergent in C series of the form F(z)=_n=1^∞a_nf(λ_n z) is proved, in particular, the following statement (Corollary 1): If condition _n=1^∞1nΓ_f(λ_n)<+∞ holds, then the relation M_F(r)=(1+o(1))μ_F(r) holds as r→ +∞ outside a set of finite logarithmic measure, where μ_F(r)={|a_n|M_f(rλ_n)\! n≥ 0}, M_F(r)={|F(z)|\! |z|=r}.