On the least possible type of entire functions of order $ \rho\in(0,1)$ with positive zeros

The problem on the minimal type of entire functions of order $\rho\in(0,1)$ with positive zeroes of prescribed densities and step

“…a parameter and we have used that 1/ → 1 as ℎ → 0. Thus, the second term in formula (3) vanishes and estimate (2) gives the sharp result by V.B. Sherstykov in work [2]:…”

“…This is why in (6) we have max instead of sup in (2). We denote by 0 = 0 ( , , ℎ, ) the point of the maximum of ( ) on the ray > 0 and we introduce the function…”

“…since the error terms in the expression for ( ) makes no influence on ( 0 ) (see [2]). Thanks to general estimate (8), in order to prove identity (9), it is sufficient to establish…”

“…Following [2], [3], we first define a sequence ( ) ∞ =1 satisfying the condition +1 = 4 , ∈ N, and we let = / ∈ (0, 1). We observe that for each > 0 the relations…”

Development of the Valiron–Levin Theorem on the Least Possible Type of Entire Functions with a Given Upper ρ-Density of Roots

“…The upper averaged 1-density of the sequence Z is equal to 2; hence, applying the first relation of (1.9), we obtain the uniqueness space [1,2). Applying the second relation of (1.9), we obtain even worse result.…”