An arXiv preprint reports a sharper version of Carlson's integral inequality. Its main result adds a squared relative-distance remainder to the continuous lower bound: J[f] is bounded below by 1/π plus 1/π times the squared distance from f to the extremal set. The extra term records how far a function is from the family that gives the minimum. The paper states that the coefficient 1/π in the refinement is optimal.
The paper asks whether this kind of remainder can sharpen Carlson's inequality while tracking relative distance to its extremal functions, and whether the same strategy works for discrete and finite-sum versions. It develops the question across nonnegative functions, infinite nonnegative sequences and finite nonnegative vectors, each within the normalization and summability conditions set for that problem.
The continuous inequality
For the continuous case, the domain is nonnegative functions in C1 with a weighted L2 condition and L1 norm equal to 1. The proof works by expanding a square around a candidate function. It chooses φ=φz to cancel the cross term, then sets z=rf. This square-completion step produces the remainder structure used in the main bound.
The extremal set is E={φz:z>0}, the family of normalized continuous minimizers. The result therefore attaches its distance term to a specific target, rather than to an unspecified reference. The paper reports that 1/π is the best possible coefficient for this refinement.
A cutoff in the discrete problem
The infinite-sequence version uses D1, the class of nonnegative sequences with finite weighted square sum and normalization to 1. The paper applies the same identity with φz and selects the scale ra, again leaving a nonnegative remainder representation. This links the discrete result to the continuous square-completion argument.
The discrete optimum is written as μ(θ)=1/ν̄θ, where ν̄θ is the maximum value of the relevant νθ. The paper finds ν̄θ=π if and only if θ lies between 0 and 1/2, inclusive.
That same point, θ=1/2, divides the attainment behavior. For θ in [0,1/2], the infimum is not attained. For θ in (1/2,1), μ(θ)<1/π, and minimizers are characterized through a nonempty set Sθ. To analyze the threshold, the paper derives νθ(z)=π+(2θ−1)z+o(z) as z approaches 0.
Finite sums keep the variational structure
In the finite-sum setting, D1,N contains nonnegative N-term vectors whose entries sum to 1, with N≥2. Here the optimal value is characterized as μN=1/ν̄N. Interior maximizers Z of νN produce the minimizers φZ, and every minimizer is obtained in this way.
The paper also gives the lower bound μN≥1/[2 arctan(N Z2)]>1/[2 arctan N], using a reported value of Z2 approximately equal to 0.811. It states that μN tends to π as N tends to infinity.
Publication and support
The document is an arXiv version 1 preprint dated 26 August 2026. The work was partially funded by the Chinese Academy of Sciences President’s International Fellowship Initiative Grant No. 2025PVA0101, and part was done during hospitality at the Innovation Academy for Precision Measurement Science and Technology in Wuhan.
Paper data and sources
Original title: A sharpened Carlson's integral inequality
Authors: Philippe Laurençot
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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