Generalized quasianalyticity and a uniqueness criterion for a class of analytic functions

In this work there is considered a class of analytic functions $\varphi(x)$ bounded on the angular sector $|\arg x|<\pi\alpha/2$, $0\leqslant\alpha<\infty$, for which
$$ \|\varphi^{(n)}\|_{L^p(0,\infty(\theta))}\leqslant m_n,\quad1\leqslant p\leqslant\infty,\quad\theta\in\biggl[-\frac{\pi\alpha}2,\frac{\pi\alpha}2\biggr], $$
such that $\varphi^{(\nu_n)}(0+)=0$, where $\{\nu_n\}$ is a subsequence of the sequence $\{n\}_{n=0}^\infty$. Under a sufficiently general assumption on $\{\nu_n\}$ a criterion is obtained for the triviality of this class, from which several known results are derived as special cases. The results are formulated in terms, introduced by the author, of derivatives of a more general form.