Constrained Lebedev–Milin Inequality

• The constrained Lebedev–Milin inequality sharpens classical estimates by imposing vanishing-moment or center-of-mass constraints to lower the optimal constant in logarithmic embeddings.
• On the circle, the method yields explicit constants like 1/[4(m+1)] through a combination of Toeplitz determinants, Fourier analysis, and variational techniques.
• Extensions to the sphere and univalent function theory demonstrate applications in Sobolev trace inequalities, coercivity estimates, and phase transition models.

The constrained Lebedev–Milin inequality is a family of sharp logarithmic-exponential estimates obtained by supplementing the classical Lebedev–Milin inequality with vanishing-moment, center-of-mass, or weighted-coefficient constraints. In its circle form, the constraint that the first $m$ moments of the exponential measure $e^{u(\theta)}\,d\theta$ vanish lowers the optimal constant from $1/4$ to $1/[4(m+1)]$. On the sphere $S^2$, an analogous sharp inequality controls the quadratic mass-moment $$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$ with threshold $\alpha_{\min}=2/3$. Related formulations occur in the theory of logarithmic coefficients of univalent functions, in sharp Sobolev trace inequalities, and in coercivity estimates for mean-field free energies (Chang et al., 2019, Chang et al., 2021, Ponnusamy et al., 2019, Case et al., 2018, Mun et al., 17 Apr 2026).

1. Circle formulation and higher-moment constraints

Let $D\subset \mathbb R^2$ be the unit disk, $S^1=\partial D$, and let $u\in H^1(D)$ satisfy $e^{u(\theta)}\,d\theta$0. The classical sharp Lebedev–Milin estimate states that

$e^{u(\theta)}\,d\theta$1

where the $e^{u(\theta)}\,d\theta$2-norm is understood as the Dirichlet energy. Equality is attained exactly on the one-parameter family

$e^{u(\theta)}\,d\theta$3

whose boundary trace has zero mean and saturates the inequality (Chang et al., 2019).

For each integer $e^{u(\theta)}\,d\theta$4, Chang–Hang introduce the constraint space

$e^{u(\theta)}\,d\theta$5

Equivalently, if $e^{u(\theta)}\,d\theta$6, then $e^{u(\theta)}\,d\theta$7 and the first $e^{u(\theta)}\,d\theta$8 complex moments of the exponential measure $e^{u(\theta)}\,d\theta$9 vanish. Their theorem states that for every integer $1/4$0 and every $1/4$1,

$1/4$2

In boundary Fourier form,

$1/4$3

whenever $1/4$4 and $1/4$5 for $1/4$6 (Chang et al., 2019).

The first cases are explicit. The case $1/4$7 gives the classical constant $1/4$8; $1/4$9 yields $1/[4(m+1)]$0, previously obtained by Osgood–Phillips–Sarnak; $1/[4(m+1)]$1 yields $1/[4(m+1)]$2. In this normalization the constants step down as

$1/[4(m+1)]$3

Thus the effect of the constraint is an explicit improvement of the energy-to-exponential-moment embedding when additional low-frequency moments of the exponential measure are removed (Chang et al., 2019).

2. Sharpness, equality, and proof mechanisms on $1/[4(m+1)]$4

The circle theory combines Toeplitz-determinant arguments, variational analysis, and Fourier expansion. In the Toeplitz-determinant approach, one considers the determinant whose symbol is $1/[4(m+1)]$5. Imposing the vanishing of the first $1/[4(m+1)]$6 moments forces the Szegő limit theorem to gain a factor $1/[4(m+1)]$7 in the exponent, which is the analytic origin of the improved constant (Chang et al., 2019).

The variational approach fixes $1/[4(m+1)]$8 and studies

$1/[4(m+1)]$9

on $S^2$0. For $S^2$1, one shows $S^2$2 by a compactness argument; at the critical value $S^2$3, the infimum is zero and is attained only by the trivial $S^2$4 in the constrained class. The Euler–Lagrange equation becomes a harmonic-extension problem in $S^2$5 with boundary condition

$S^2$6

Using the orthogonality $S^2$7 and a careful Fourier expansion, all Lagrange multipliers $S^2$8 are shown to vanish. The remaining ODE admits only the classical one-pole solutions

$S^2$9

which violate the vanishing-moment conditions unless $$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$0, so $$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$1 is constant (Chang et al., 2019).

A closely related exact statement appears in the periodic formulation on

$$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$2

If $$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$3 is real-valued, $$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$4 is its Poisson–harmonic extension to the unit disk, and

$$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$5

then

$$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$6

Here the constant $$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$7 is sharp, and equality holds if and only if $$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$8 is a trigonometric polynomial of degree at most $$\bigl(\int e^{2u}\bigr)^2-\bigl|\int x\,e^{2u}\bigr|^2$$9, equivalently a translate of the Fejér–Rogosinski extremizer (Mun et al., 17 Apr 2026).

3. Sphere analogue and the center-of-mass term

On the round sphere $\alpha_{\min}=2/3$0 with normalized area form $\alpha_{\min}=2/3$1, Chang–Gui define

$\alpha_{\min}=2/3$2

and for $\alpha_{\min}=2/3$3,

$\alpha_{\min}=2/3$4

Their constrained Lebedev–Milin inequality asserts that the sharp threshold is

$\alpha_{\min}=2/3$5

Precisely, for every $\alpha_{\min}=2/3$6,

$\alpha_{\min}=2/3$7

equivalently,

$\alpha_{\min}=2/3$8

while for any $\alpha_{\min}=2/3$9 one can find $D\subset \mathbb R^2$0 with $D\subset \mathbb R^2$1 (Chang et al., 2021).

In the special case $D\subset \mathbb R^2$2, the inequality reduces to the mass-centered Aubin–Onofri setting. The circle relation is explicit in the source: Chang–Gui’s theorem is in exact analogy to Szegő’s second inequality on $D\subset \mathbb R^2$3, with the center-of-mass deviation replacing the first Fourier moment. The sphere statement therefore quantifies how a nonzero center of mass degrades the Onofri bound, while preserving sharpness (Chang et al., 2021).

The proof is variational. One studies critical points under the two constraints

$D\subset \mathbb R^2$4

which produce the Euler–Lagrange system

$D\subset \mathbb R^2$5

A Kazdan–Warner identity forces $D\subset \mathbb R^2$6 unless $D\subset \mathbb R^2$7. At $D\subset \mathbb R^2$8, stereographic projection reduces the problem to a Liouville equation in $D\subset \mathbb R^2$9 with an explicit radially symmetric solution, and one proves that the infimum of $S^1=\partial D$0 on all of $S^1=\partial D$1 is zero for $S^1=\partial D$2 and is unattained below that threshold (Chang et al., 2021).

Equality is attained exactly by the family of conformal factors arising from Möbius automorphisms of $S^1=\partial D$3. In stereographic coordinates one may write the unique critical solution with center-of-mass $S^1=\partial D$4 as

$S^1=\partial D$5

The zero-mode and first-harmonic spherical harmonics form the kernel of the linearized second variation at these critical points (Chang et al., 2021).

4. Cubature, spherical designs, and near-extremals

A structural feature of the constrained inequalities is that the improved constant is governed by finite moment-annihilating configurations. On $S^1=\partial D$6, if one imposes vanishing of all area-moments up to degree $S^1=\partial D$7, the optimal constant becomes $S^1=\partial D$8, where $S^1=\partial D$9 is the minimal size of a positive-weight cubature of degree $u\in H^1(D)$0 on $u\in H^1(D)$1. In the circle case one has

$u\in H^1(D)$2

which recovers the factor $u\in H^1(D)$3 (Chang et al., 2019).

The circle moment condition may also be written polynomially. Let $u\in H^1(D)$4 be the unit disk, $u\in H^1(D)$5, $u\in H^1(D)$6 the real polynomials on $u\in H^1(D)$7 of total degree $u\in H^1(D)$8, and

$u\in H^1(D)$9

If $e^{u(\theta)}\,d\theta$00, $e^{u(\theta)}\,d\theta$01 is an extension of $e^{u(\theta)}\,d\theta$02, and

$e^{u(\theta)}\,d\theta$03

then

$e^{u(\theta)}\,d\theta$04

and the sharp constant is $e^{u(\theta)}\,d\theta$05 (Chen et al., 2021).

The sharpness mechanism is a bubble construction indexed by a design. One chooses $e^{u(\theta)}\,d\theta$06 nodes $e^{u(\theta)}\,d\theta$07 and positive weights $e^{u(\theta)}\,d\theta$08 with $e^{u(\theta)}\,d\theta$09 such that

$e^{u(\theta)}\,d\theta$10

Local bubbles are then placed in disjoint conic neighborhoods: $e^{u(\theta)}\,d\theta$11 and glued by cutoffs into a global test function

$e^{u(\theta)}\,d\theta$12

For its trace $e^{u(\theta)}\,d\theta$13, the moment constraints are exact, and the asymptotics are

$e^{u(\theta)}\,d\theta$14

Hence

$e^{u(\theta)}\,d\theta$15

so no smaller constant can hold (Chen et al., 2021).

5. Weighted Milin inequalities for logarithmic coefficients

The phrase “constrained Lebedev–Milin inequality” also appears in univalent function theory. Let

$e^{u(\theta)}\,d\theta$16

be analytic on the unit disk, and define the logarithmic coefficients by

$e^{u(\theta)}\,d\theta$17

For $e^{u(\theta)}\,d\theta$18 univalent, de Branges’ theorem yields, for every integer $e^{u(\theta)}\,d\theta$19,

$e^{u(\theta)}\,d\theta$20

with equality for all $e^{u(\theta)}\,d\theta$21 simultaneously if and only if $e^{u(\theta)}\,d\theta$22 is the Koebe function $e^{u(\theta)}\,d\theta$23 or one of its rotations (Ponnusamy et al., 2019).

A general weighted version is obtained by replacing the coefficient $e^{u(\theta)}\,d\theta$24 with a convex weight sequence. If $e^{u(\theta)}\,d\theta$25 is a convex sequence of nonnegative real numbers satisfying

$e^{u(\theta)}\,d\theta$26

then for every $e^{u(\theta)}\,d\theta$27,

$e^{u(\theta)}\,d\theta$28

Equality holds if and only if $e^{u(\theta)}\,d\theta$29 for some $e^{u(\theta)}\,d\theta$30 (Ponnusamy et al., 2019).

The proof rewrites the convexity condition through

$e^{u(\theta)}\,d\theta$31

together with

$e^{u(\theta)}\,d\theta$32

and then sums the classical Milin inequalities against $e^{u(\theta)}\,d\theta$33. This is a summation-by-parts reduction from the weighted statement to de Branges’ finite inequalities (Ponnusamy et al., 2019).

Roth’s extension relaxes global convexity. If $e^{u(\theta)}\,d\theta$34 for all $e^{u(\theta)}\,d\theta$35, one studies a finite weighted inequality through de Branges’ ODE system

$e^{u(\theta)}\,d\theta$36

and positivity of certain Jacobi-polynomial combinations

$e^{u(\theta)}\,d\theta$37

This permits sharp nonconvex examples, including

$e^{u(\theta)}\,d\theta$38

and, for $e^{u(\theta)}\,d\theta$39,

$e^{u(\theta)}\,d\theta$40

(Ponnusamy et al., 2019).

6. Higher-order analogues and applications

The constrained Lebedev–Milin framework extends beyond second-order boundary analysis. For a compact manifold with boundary $e^{u(\theta)}\,d\theta$41, Case and Luo construct a sixth-order GJMS operator $e^{u(\theta)}\,d\theta$42, associated conformally covariant boundary operators $e^{u(\theta)}\,d\theta$43, and the energy

$e^{u(\theta)}\,d\theta$44

On compactifications of Poincaré–Einstein manifolds, the boundary operators realize fractional GJMS operators of orders one, three, and five. In critical dimension $e^{u(\theta)}\,d\theta$45, the sharp Onofri–Lebedev–Milin inequality on $e^{u(\theta)}\,d\theta$46,

$e^{u(\theta)}\,d\theta$47

yields the six-dimensional Lebedev–Milin or Onofri-type trace inequality

$e^{u(\theta)}\,d\theta$48

where $e^{u(\theta)}\,d\theta$49, $e^{u(\theta)}\,d\theta$50, and $e^{u(\theta)}\,d\theta$51. Equality holds precisely when $e^{u(\theta)}\,d\theta$52 and each boundary datum has the Möbius-invariant extremal form recorded in the source (Case et al., 2018).

A different application appears in mean-field free energies on the circle. Proposition 2.1 in the 2026 work on phase transitions uses the constrained Lebedev–Milin inequality in the form

$e^{u(\theta)}\,d\theta$53

under vanishing of the first $e^{u(\theta)}\,d\theta$54 nonzero Fourier modes of $e^{u(\theta)}\,d\theta$55, to obtain a sharp coercivity estimate. Combined with the dual form of the inequality and the Donsker–Varadhan representation of relative entropy, this shows that for an $e^{u(\theta)}\,d\theta$56-periodic interaction satisfying

$e^{u(\theta)}\,d\theta$57

the uniform distribution remains the unique minimizer for $e^{u(\theta)}\,d\theta$58, and the transition occurs exactly at $e^{u(\theta)}\,d\theta$59 (Mun et al., 17 Apr 2026).

The same source applies this coercive mechanism to several models for which the exact value of the phase transition and its continuity were not fully known. For the two-dimensional Doi–Onsager model $e^{u(\theta)}\,d\theta$60, the phase transition is continuous at

$e^{u(\theta)}\,d\theta$61

For the noisy transformer model $e^{u(\theta)}\,d\theta$62, a sharp threshold $e^{u(\theta)}\,d\theta$63 is identified such that $e^{u(\theta)}\,d\theta$64 and the phase transition is continuous for $e^{u(\theta)}\,d\theta$65, while $e^{u(\theta)}\,d\theta$66 and the phase transition is discontinuous for $e^{u(\theta)}\,d\theta$67. An analogous sharp dichotomy is obtained for the noisy Hegselmann–Krause model $e^{u(\theta)}\,d\theta$68 (Mun et al., 17 Apr 2026).

Across these formulations, the common mechanism is the removal or controlled weighting of low modes: Fourier moments of the exponential measure on $e^{u(\theta)}\,d\theta$69, center-of-mass modes on $e^{u(\theta)}\,d\theta$70, or coefficient modes in the logarithmic expansion of a univalent function. The constrained inequality is therefore not a single formula but a sharp principle linking moment annihilation, extremal conformal factors, Toeplitz or de Branges structures, and optimal constants in critical logarithmic embeddings (Chang et al., 2019, Chang et al., 2021, Ponnusamy et al., 2019).