Geodesic
An example of a fast decaying $(p,A)$-lacunary function
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 350-363

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A function $f(z)=\sum^\infty_{k=0}a_kz^{n_k}$ analytic in the unit disc is said be $(p,A)$-lacunary if $n_k\ge Ak^p$, $1$, $A>0$ for all $k\ge0$. For $1$, $A>0$, a $(p,A)$-lacunary function $f_{1,p,A}(z)$ is constructed that decays as $x\to1-0$ almost optimally. 
@article{ZNSL_2000_270_a18,
     author = {N. A. Shirokov},
     title = {An example of a fast decaying $(p,A)$-lacunary function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {350--363},
     publisher = {mathdoc},
     volume = {270},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a18/}
}
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N. A. Shirokov. An example of a fast decaying $(p,A)$-lacunary function. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 28, Tome 270 (2000), pp. 350-363. http://geodesic.mathdoc.fr/item/ZNSL_2000_270_a18/