Geodesic
On the decay rate of $(p,A)$-lacunary series
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 135-149 
F. L. Nazarov; N. A. Shirokov. On the decay rate of $(p,A)$-lacunary series. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 135-149. http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a8/
@article{ZNSL_2005_327_a8,
     author = {F. L. Nazarov and N. A. Shirokov},
     title = {On the decay rate of $(p,A)$-lacunary series},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {135--149},
     year = {2005},
     volume = {327},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a8/}
}
TY  - JOUR
AU  - F. L. Nazarov
AU  - N. A. Shirokov
TI  - On the decay rate of $(p,A)$-lacunary series
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2005
SP  - 135
EP  - 149
VL  - 327
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a8/
LA  - ru
ID  - ZNSL_2005_327_a8
ER  - 
%0 Journal Article
%A F. L. Nazarov
%A N. A. Shirokov
%T On the decay rate of $(p,A)$-lacunary series
%J Zapiski Nauchnykh Seminarov POMI
%D 2005
%P 135-149
%V 327
%U http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a8/
%G ru
%F ZNSL_2005_327_a8

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A power series $\sum\limits^\infty_{k=0} a_k x^{n_k}$ with radius of convergence equal to 1 is said to be $(p,A)$-lacunary if $n_k\ge Ak^p$, $A>0$, $1. It is proved that if a $(p,A)$-lacunary series $f$ satisfies the condition $$ |f(x)|\exp\biggl(B(1-x)^{-\frac1{p-1}}+\varepsilon(1-x)^{-\frac1{p-1}}\bigg/(|\log(1-x)|+1)\biggr)\underset{x\to1-0}{\longrightarrow}0, $$ for $1, where $$ B=(p-1)\biggl(\frac\pi p\biggr)^{\frac p{p-1}}\cdot\frac1{A^{1/(p-1)}}\cdot\frac1{|\cos\frac{\pi p}2|^{1/(p-1)}}, $$ and $\varepsilon>0$, then $f\equiv0$. We also construct a $(p,A)$-lacunary series $f_0$ such that $$ |f_0(x)|\exp\biggl(B(1-x)^{-\frac1{p-1}}+C_0(1-x)^{-\frac1{p-1}}\bigg/(|\log(1-x)|^2+1)\biggr)\underset{x\to1-0}{\longrightarrow}0. $$ for a constant $C_0=C_0(p,A)>0$.

[1] N. A. Shirokov, Analytic functions smooth up to the boundary, Lect. Notes Math., 1312, 1988 | MR | Zbl

[2] N. A. Shirokov, “Primer bystro ubyvayuschei $(p,A)$-lakunarnoi funktsii”, Zap. nauchn. semin. POMI, 270, 2000, 350–363 | MR | Zbl

[3] L. I. Hirschman, J. A. Jenkins, “On lacunary Dirichlet series”, Proc. Amer. Math. Soc, 1:4 (1950), 512–517 | DOI | MR | Zbl

[4] J. M. Anderson, “Bounded analytic functions with Hadamard gaps”, Mathematica, 23:2 (1976), 142–147 | MR