Fourier series and $\delta$-subharmonic functions of finite $\gamma$-type in a half-plane

Let $\gamma(r)$ be a growth function and let $v(z)$ be a proper $\delta$-subharmonic function in the sense of Grishin in a complex half-plane, that is $v=v_1-v_2$, where $v_1$ and $v_2$ are proper subharmonic functions $(\lim\sup_{z\to t}v_i(z)\leqslant0$, for each real $t$, $i=1,2)$, let $\lambda=\lambda_+-\lambda_-$ be the full measure corresponding to $v$ and let $T(r,v)$ be its Nevanlinna characteristic. The class $J\delta(\gamma)$ of functions of finite $\gamma$-type is defined as follows: $v\in J\delta(\gamma)$ if $T(r,v)\leqslant A\gamma(Br)/r$ for some positive constants $A$ and $B$. The Fourier coefficients of $v$ are defined in the standard way:
$$ c_k(r,v)=\frac 2\pi\int_0^\pi v(re^{i\theta})\sin k\theta\,d\theta, \qquad r>0, \quad k\in\mathbb N. $$

The central result of the paper is the equivalence of the following properties:

• (1) $v\in J\delta(\gamma)$;
• (2) $N(r)\leqslant A_1\gamma(B_1r)/r$,

where $N(r)=N(r,\lambda_+)$ or $N(r)=N(r,\lambda_-)$, and $|c_k(r,v)|\leqslant A_2\gamma(B_2r)$. It is proved in addition that $J\delta(\gamma)=JS(\gamma)-JS(\gamma)$, where $JS(\gamma)$ is the class of proper subharmonic functions of finite $\gamma$-type.