The indicator of a delta-subharmonic function in a half-plane

Delta-subharmonic functions of a completely regular growth in the upper half-plane have been introduced in the joint work of the authors, published in Reports of the Russian Academy of Sciences (2001). In this work, criteria whether a delta-subharmonic function in the upper half-plane belongs to a class of functions of a completely regular growth have been obtained on the basis of the theory of Fourier coefficients of delta-subharmonic functions in the half-plane developed in the beginning of this century by the first author of the present article. The present paper is a natural continuation of this research. The concept of the indicator of a delta-subharmonic function of a completely regular growth in the upper half-plane is introduced. It is proved that the indicator of a delta-subharmonic function of a completely regular growth in the upper half-plane belongs to a class $L_p[0,\pi]$ ($1<p\leq2$). The proof is based on the lemma about Polya peaks and the Hausdorff–Young theorem.