A spread relation for entire functions with negative zeros

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Let g g be a canonical product having only real negative zeros and nonintegral order λ \lambda , and let ϕ \phi be the set function defined by 2 π ϕ ( E ) = ∫ E π λ csc ⁡ π λ cos ⁡ λ θ d θ 2\pi \phi (E) = {\smallint _E}\pi \lambda \csc \pi \lambda \cos \lambda \theta d\theta . It is shown that if E ( r ) E(r) is the set of values of θ ∈ ( − π , π ] \theta \in ( - \pi ,\pi ] where | g ( r e i θ ) | ≥ 1 , r n |g(r{e^{i\theta }})| \geq 1,{r_n} is a sequence of Polya peaks of g g and δ \delta is the deficiency of the value zero of g g then ϕ ( E ( r n ) ) ≥ 2 ( 1 − δ ) − 1 \phi (E({r_n})) \geq 2{(1 - \delta )^{ - 1}} . This inequality leads to a sharp spread relation for g g .