Let $G$ be an arbitrary domain of the complex plane. Denote by $\mathcal H(G)$ the space of all analytic functions in $G$. Let $G_{1}$, $G_{2}$ be arbitrary domains of the complex plane. A linear operator $A:\mathcal H(G_{1}) \longrightarrow \mathcal H(G_{2})$ is called multiplicative operator if $A(fg)=A(f)A(g)$ for any $f,g \in \mathcal H(G_{1})$. In this paper we study solutions of operator equations which are modifications of the multiplicative equation and contain operators of multiplication by analytic functions. We investigate conditions for the existence of nonzero linear operators $A:\mathcal H(G_{1}) \longrightarrow \mathcal H(G_{2})$ such that $\varphi(z)(A(fg))(z)=\varphi_{1}(z)(Af)(z)(Ag)(z)$, where $f,g \in \mathcal H(G_{1})$, $z\in G_{2}$, $\varphi$, $\varphi_{1}$ are arbitrary fixed functions of $\mathcal H(G_{2})$ that is not identically equal to zero in $G_{2}$. In case that there exist nontrivial solutions we describe all solutions the above equation in the class of of linear operators $A:\mathcal H(G_{1}) \longrightarrow \mathcal H(G_{2})$. We describe all ordered sets of linear operators which act from one space of analytic functions in an arbitrary domain of the complex plane into other analogous space and satisfy generalized multiplicative operator relation for an arbitrary finite natural number of multipliers. We obtain the description of all linear operators $A,B_{j}: \mathcal H(G_{1}) \longrightarrow \mathcal H(G_{2})$ such that $A\left(\prod \limits_{j=1}^{n} f_{j}\right)= \prod \limits_{j=1}^{n} B_{j}(f_{j})$ for all $f_{j} \in \mathcal H(G_{1})$, $j=\overline{1,n}$. To solve the above mentioned operator equations we use the method of reduction these equations to the respective equations in the class of linear functionals on the space $\mathcal H(G_{1})$ and extension theorems for analytic functions.

[1] Nandakumar N.R. Ring homomorphisms on H(G). Int. J. Math. Math. Sci. 1990, 13 (2), 393-396. doi:10.1155/S016117129000059X

[2] Nandakumar N.R. Ring homomorphisms on algebras of analytic functions. Ann. Univ. Mariae Curie-Sklodowska, Sect A. 1990, 44, 37-43.

[3] Garnett J.B. Bounded analytic functions, Graduate texts in mathematics 236. Springer-Verlag, New York, 2007. doi:10.1007/0-387-49763-3

[4] Linchuk Yu.S. On a generalization of Rubel's equation. Int. J. Math. Anal. 2015, 9 (5), 231-236. doi:10.12988/ijma.2015.412405

[5] Linchuk Yu.S., Linchuk S.S. On generalized Rubel's equation. Aeq. Math. 2017, 91 (3), 537-545. doi:10.1007/s00010-017-0467-x

[6] Linchuk Yu.S. On a generalized Rubel's operator equation. Miskolc. Math. Notes. 2017, 17 (2), 925-930. doi:10.18514/MMN.2017.1526

[7] Linchuk Yu.S. On Rubel type operator equations on the space of analytic functions. Rend. Circ. Mat. Palermo (2). 2017, 66 (3), 383-389. doi:10.1007/s12215-016-0263-9