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The two-sided estimate

Abstract

Hyperbolic bounds for the Kilbas-Saigo function on the negative half-line.

Theorem 1.1 (The full parameter range).

Lean statement: D5/S3/Analytic/KilbasSaigo/Result.result

Proof. Machine-checked in Lean as D5/S3/Analytic/KilbasSaigo/Result.result (✓ std3). ∎

Resolves. Problems/boudabsa-simon-2021-kilbas-saigo-hyperbolic-bounds (proved) by D5/S3/Analytic/KilbasSaigo/Result.result.

Source. Repository-derived.

Acknowledgement. L. Boudabsa and T. Simon (2021). Some properties of the Kilbas–Saigo function. DOI: 10.3390/math9030217. URL: https://arxiv.org/abs/2012.05666.

Commentary.

For every real alpha,m,l,x with 0<alpha<=1, m>0, l>m-1/alpha and x>=0, E(alpha,m,l)(-x) is bounded below by 1/(1+Gamma(1+alpha(l-m)) x/Gamma(1+alpha(l-m+1))) and above by 1/(1+Gamma(1+alpha l) x/Gamma(1+alpha(1+l))). The defining series and its derivatives converge on the entire real line. Positive Gamma increment kernels establish the fractional range; affine differential equations establish alpha=1. Strict positivity, decreasingness and convexity give the two estimates.

References