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The derivative sign theorem

Abstract

Derivative signs of the power-weighted hyperbolic cotangent remainder.

Theorem 1.1 (The two derivative orders).

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

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

Resolves. Problems/zhu-2020-coth-bernoulli-remainder-derivative-signs (proved) by D5/S3/Analytic/CothBernoulliDegree/Result.result.

Source. Repository-derived.

Acknowledgement. L. Zhu (2020). Completely monotonic integer degrees for a class of special functions. DOI: 10.3934/math.2020224. URL: https://www.aimspress.com/article/doi/10.3934/math.2020224.

Commentary.

Let m and n be natural numbers with m at least one and n at least two, and put M=m+2n. The signed remainder P has derivative of order M-2 strictly positive at every positive t. Its derivative of order M-1 fails to be strictly positive everywhere on the positive ray.

The partial-fraction formula and its differentiated series express these derivatives as positive multiples of rational-kernel derivatives. At order M-2 each kernel is strictly positive. The derivative series is summable, so its sum is strictly positive.

At t=5 pi/M, the derivative of order M-1 is at most -(M-1)!/(8 pi^(2n+1)). The first normalized summand is at most -1/8 by the phase and amplitude estimate. The remaining normalized summands are bounded above by a reciprocal-power tail of at most 1/16. Restoring the common positive factor gives the stated strictly negative value and hence the second clause.

References