Delta-subharmonic functions of completely regular gamma-growth in a half-plane

We consider entire, analytic, and delta-subharmonic functions of completely regular growth relative to the proximate order in a half-plane and to an arbitrary function of growth γ ( r ) , which satisfies to the inequality γ (2 r ) ≤ Kγ ( r ) at some K > 0 and all r > 0 . The concept of indicator h ( θ, v ) of a delta-subharmonic function v in a half-plane relative to γ ( r ) is introduced. It is proved that the indicator h ( θ, v ) belongs to the space L 2 [0 , π ].