Estimate of a polynomial in generalized exponential functions in the convex hull of two domains (Q1114035)

This is related to extensive work by A. F. Leont'ev. Consider the generalized exponential functions \[ f_ n(z)=(1+\psi_ n(z))\exp \lambda_ nz,\quad n=1,..., \] holomorphic in a simply connected domain S, where \(\lambda_ 1,\lambda_ 2,..\). are zeros of an entire function L(\(\lambda)\) of exponential type (subject to certain conditions). Let \(\bar {\mathcal D}\) denote the conjugate indicator diagram of L and put \(\bar {\mathcal D}_{\alpha}=\bar {\mathcal D}+\alpha\), \(\alpha\in C\). Assume that the \(f_ n\) satisfy the uniqueness condition: \(\sum^{\infty}_{n=1}b_ nf_ n(z)=0\), \(\forall z\in E\), \(\bar {\mathcal D}_{\alpha}\subset E\subset S\), implies \(\forall b_ n=0\). Assume also, concerning the \(\psi_ n\), that for each compact \(K\subset S\), \(\exists A(K)\), q(K), \(0<q(K)<1\), such that \[ | \psi_ n(z)| \leq A q^{| \lambda_ n|},\quad \forall z\in K,\quad \forall n\geq 1. \] It is shown that if a domain E is such that \(\bar {\mathcal D}_{\alpha}\subset E\), \(\bar E\subset S\), then there exists a domain G and a constant B such that \(\bar {\mathcal D}_{\alpha}\subset G\), \(\bar G\subset E\) and \[ \sum^{n}_{\nu =1}| a_{\nu} \exp \lambda_{\nu}z| \leq B\max_{t\in E}| \sum^{n}_{\nu =1}a_{\nu}f_{\nu}(t)|,\quad \forall z\in G,\quad \forall n\geq 1, \] B being independent of the sequence \(\{a_{\nu}\}\). A similar estimate involving the convex hull of two domains \({\mathcal D}_ 1,{\mathcal D}_ 2\subset S\) is also proven.