To the boundedness of the \(l-\mathfrak{M}\) -index of functions analytic in the unit disk and represented by series in systems of functions

<p>Let <span class="math inline">\(f(z) = \sum\limits_{k=0}^{\infty} f_k z^k\)</span> be an entire transcendental function, and let <span class="math inline">\((\lambda_n)\)</span> be a sequence of positive numbers increasing to <span class="math inline">\(+\infty\)</span>. Suppose that the series <span class="math inline">\(A(z) = \sum\limits_{n=1}^{\infty} a_n f(\lambda_n z)\)</span> is regularly convergent in <span class="math inline">\(\mathbb{D} = \{ z : |z| < 1 \}\)</span>, i.e., <span class="math inline">\(\mathfrak{M}(r, A) := \sum\limits_{n=1}^{\infty} |a_n| M_f(r \lambda_n) < +\infty\)</span> for all <span class="math inline">\(r \in [0, 1)\)</span>. For a positive function <span class="math inline">\(l\)</span> continuous on <span class="math inline">\([0, 1)\)</span>, the function <span class="math inline">\(A\)</span> is said to be of bounded <span class="math inline">\(l\)</span>-<span class="math inline">\(\mathfrak{M}\)</span>-index if there exists <span class="math inline">\(N \in \mathbb{Z}_+\)</span> such that <span class="math display">\[\frac{\mathfrak{M}(r, A^{(n)})}{n! \, l^n(r)} \leq \max \left\{ \frac{\mathfrak{M}(r, A^{(k)})}{k! \, l^k(r)} : 0 \leq k \leq N \right\},\]</span> for all <span class="math inline">\(n \in \mathbb{Z}_+\)</span> and all <span class="math inline">\(r \in [0, 1)\)</span>. The growth of bounded <span class="math inline">\(l\)</span>-<span class="math inline">\(\mathfrak{M}\)</span>-index functions is studied. In particular, under the conditions <span class="math inline">\(a_n \geq 0\)</span> and <span class="math inline">\(f_k \geq 0\)</span>, it is proved that the function <span class="math inline">\(A\)</span> is of bounded <span class="math inline">\(l\)</span>-<span class="math inline">\(\mathfrak{M}\)</span>-index with <span class="math inline">\(l(r) = p(1 - r)^{-(p+1)}\)</span>, <span class="math inline">\(p > 0\)</span>, if and only if <span class="math display">\[\lim\limits_{r \uparrow 1} (1 - r)^p \ln \mathfrak{M}(r, A) < +\infty.\]</span> This condition is satisfied if and only if <span class="math display">\[\lim\limits_{k \to \infty} k^{-p} \left( \ln^+ (f_k \mu_D(k)) \right)^{p+1} < +\infty,\]</span> where <span class="math inline">\(\mu_D(k) = \max\{ a_n \lambda_n^k : n \geq 1 \}\)</span>.</p>