---
title: Constrained Lebedev–Milin Inequality
url: https://www.emergentmind.com/topics/constrained-lebedev-milin-inequality
type: topic
---

# Constrained Lebedev–Milin Inequality

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 [1909.00431] [2109.13385] [1903.09974] [1810.08027] [2604.16288].

## 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 \(\int_{S^1}u\,d\theta=0\). The classical sharp Lebedev–Milin estimate states that
\[
\log\!\Bigl(\frac12\int_{S^1} e^{u(\theta)}\,d\theta\Bigr)
\le
\frac14\int_D |\nabla u|^2\,dx,
\]
where the \(H^1(D)\)-norm is understood as the Dirichlet energy. Equality is attained exactly on the one-parameter family
\[
u(z)=\log |1-a z|^{-2},\qquad |a|<1,
\]
whose boundary trace has zero mean and saturates the inequality [1909.00431].

For each integer \(m\ge 1\), Chang–Hang introduce the constraint space
\[
X_m=
\Bigl\{
u\in H^1(D):
\int_{S^1}u=0,\;
\int_{S^1} e^{u(\theta)}e^{ik\theta}\,d\theta=0
\text{ for }k=1,2,\dots,m
\Bigr\}.
\]
Equivalently, if \(f(\theta)=u(e^{i\theta})\), then \(\hat f(0)=0\) and the first \(m\) complex moments of the exponential measure \(e^f\,d\theta\) vanish. Their theorem states that for every integer \(m\ge 0\) and every \(u\in X_m\),
\[
\log\!\Bigl(\frac12\int_{S^1} e^{u(\theta)}\,d\theta\Bigr)
\le
\frac{1}{4(m+1)}\int_D |\nabla u|^2\,dx.
\]
In boundary Fourier form,
\[
\log\!\Bigl(\frac1{2\pi}\int_0^{2\pi}e^{f(\theta)}\,d\theta\Bigr)
\le
\frac1{4(m+1)}\sum_{k=1}^{\infty} k\,|\hat f(k)|^2
\]
whenever \(\hat f(0)=0\) and \(\int e^f e^{ik\theta}=0\) for \(1\le k\le m\) [1909.00431].

The first cases are explicit. The case \(m=0\) gives the classical constant \(1/4\); \(m=1\) yields \(1/8\), previously obtained by Osgood–Phillips–Sarnak; \(m=2\) yields \(1/12\). In this normalization the constants step down as
\[
\frac14\to \frac18\to \frac1{12}\to \frac1{16}\to \cdots = \frac1{4(m+1)}.
\]
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 [1909.00431].

## 2. Sharpness, equality, and proof mechanisms on \(S^1\)

The circle theory combines Toeplitz-determinant arguments, variational analysis, and Fourier expansion. In the Toeplitz-determinant approach, one considers the determinant whose symbol is \(e^{u(\theta)}\). Imposing the vanishing of the first \(m\) moments forces the Szegő limit theorem to gain a factor \(1/(m+1)\) in the exponent, which is the analytic origin of the improved constant [1909.00431].

The variational approach fixes \(a>0\) and studies
\[
J_a(u)=a\int_D |\nabla u|^2-\log\int_{S^1} e^u
\]
on \(X_m\). For \(a>1/[4(m+1)]\), one shows \(J_a\ge 0\) by a compactness argument; at the critical value \(a=1/[4(m+1)]\), the infimum is zero and is attained only by the trivial \(u\equiv 0\) in the constrained class. The Euler–Lagrange equation becomes a harmonic-extension problem in \(D\) with boundary condition
\[
\partial_n v+\frac1{4a}
=
e^v+\sum_{k=1}^m \bigl(c_k e^{ik\theta}+\overline{c_k}e^{-ik\theta}\bigr)e^v.
\]
Using the orthogonality \(\int e^v e^{ik\theta}=0\) and a careful Fourier expansion, all Lagrange multipliers \(c_k\) are shown to vanish. The remaining ODE admits only the classical one-pole solutions
\[
v(z)=\log\!\Bigl[\frac{(m+1)(1-|\alpha|^2)}{|1-\alpha z^{m+1}|^2}\Bigr],
\]
which violate the vanishing-moment conditions unless \(\alpha=0\), so \(v\) is constant [1909.00431].

A closely related exact statement appears in the periodic formulation on
\[
T=[-\tfrac12,\tfrac12).
\]
If \(\phi\in L^1(T)\) is real-valued, \(\Phi\) is its Poisson–harmonic extension to the unit disk, and
\[
\int_T e^{\phi(\theta)}e^{2\pi i k\theta}\,d\theta=0
\qquad \forall\,1\le k\le n,
\]
then
\[
\log\!\Bigl(\int_T e^{\phi(\theta)}\,d\theta\Bigr)-\int_T \phi(\theta)\,d\theta
\le
\frac{1}{4(n+1)\pi}\int_D |\nabla \Phi(x,y)|^2\,dx\,dy.
\]
Here the constant \(1/[4(n+1)\pi]\) is sharp, and equality holds if and only if \(e^{-\phi(\theta)}\) is a trigonometric polynomial of degree at most \(n+1\), equivalently a translate of the Fejér–Rogosinski extremizer [2604.16288].

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

On the round sphere \(S^2\) with normalized area form \(d\omega\), Chang–Gui define
\[
M_0(u)=\int_{S^2} e^{2u}d\omega,
\qquad
\mathbf M_1(u)=\int_{S^2} x\,e^{2u(x)}\,d\omega(x)\in \mathbb R^3,
\]
and for \(\alpha>0\),
\[
I_\alpha(u)
=
\alpha\int_{S^2} |\nabla u|^2\,d\omega
+
2\int_{S^2} u\,d\omega
-
\frac12\ln\!\Bigl[M_0(u)^2-|\mathbf M_1(u)|^2\Bigr].
\]
Their constrained Lebedev–Milin inequality asserts that the sharp threshold is
\[
\alpha_{\min}=\frac23.
\]
Precisely, for every \(u\in H^1(S^2)\),
\[
I_{2/3}(u)\ge 0,
\]
equivalently,
\[
\Bigl(\int_{S^2}e^{2u}\Bigr)^2
-
\Bigl|\int_{S^2}x\,e^{2u}\Bigr|^2
\le
\exp\!\Bigl\{
\frac43\int_{S^2}|\nabla u|^2+4\int_{S^2}u
\Bigr\},
\]
while for any \(\alpha<2/3\) one can find \(u\) with \(I_\alpha(u)\to -\infty\) [2109.13385].

In the special case \(\mathbf M_1(u)=0\), 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 \(S^1\), 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 [2109.13385].

The proof is variational. One studies critical points under the two constraints
\[
\int e^{2u}=1,\qquad \int x_i e^{2u}=a_i,
\]
which produce the Euler–Lagrange system
\[
\alpha\,\Delta u+\frac{1-\sum a_i x_i}{1-|\mathbf a|^2}e^{2u}-1=0.
\]
A Kazdan–Warner identity forces \(\mathbf a=(0,0,0)\) unless \(\alpha=2/3\). At \(\alpha=2/3\), stereographic projection reduces the problem to a Liouville equation in \(\mathbb R^2\) with an explicit radially symmetric solution, and one proves that the infimum of \(I_\alpha\) on all of \(H^1(S^2)\) is zero for \(\alpha\ge 2/3\) and is unattained below that threshold [2109.13385].

Equality is attained exactly by the family of conformal factors arising from Möbius automorphisms of \(S^2\). In stereographic coordinates one may write the unique critical solution with center-of-mass \(\mathbf a\) as
\[
u_{\mathbf a}(x)
=
-\frac32\ln(1-\mathbf a\cdot x)
+
\frac12\ln\frac{2}{1-|\mathbf a|^2}.
\]
The zero-mode and first-harmonic spherical harmonics form the kernel of the linearized second variation at these critical points [2109.13385].

## 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 \(S^2\), if one imposes vanishing of all area-moments up to degree \(m\), the optimal constant becomes \(1/(4N_m)\), where \(N_m\) is the minimal size of a positive-weight cubature of degree \(m\) on \(S^2\). In the circle case one has
\[
N_m(S^1)=m+1,
\]
which recovers the factor \(1/[4(m+1)]\) [1909.00431].

The circle moment condition may also be written polynomially. Let \(B^2\) be the unit disk, \(d\mu=|d\theta|/(2\pi)\), \(P_m\) the real polynomials on \(\mathbb R^2\) of total degree \(\le m\), and
\[
\mathbb P_m^\circ=\{p\in P_m:\int_{S^1}p(\theta)\,d\mu(\theta)=0\}.
\]
If \(f\in C^\infty(S^1)\), \(u\in H^1(B^2)\) is an extension of \(f\), and
\[
\int_{S^1} e^{f(\theta)}p(\theta)\,d\mu(\theta)=0
\qquad \forall\,p\in \mathbb P_m^\circ,
\]
then
\[
\log\!\Bigl(\frac1{2\pi}\int_{S^1} e^{f(\theta)-\bar f}\,d\sigma(\theta)\Bigr)
\le
\frac1{4\pi(m+1)}\int_{B^2} |\nabla u(x)|^2\,dx,
\]
and the sharp constant is \(1/[4\pi(m+1)]\) [2107.08647].

The sharpness mechanism is a bubble construction indexed by a design. One chooses \(N=m+1\) nodes \(\theta_1,\dots,\theta_N\) and positive weights \(\nu_1,\dots,\nu_N\) with \(\sum \nu_i=1\) such that
\[
\sum_{i=1}^N \nu_i p(\theta_i)=0
\qquad \forall\,p\in \mathbb P_m^\circ.
\]
Local bubbles are then placed in disjoint conic neighborhoods:
\[
\phi_{\varepsilon,i}(r,\theta)
=
-\log\bigl(\varepsilon^2+(1-r)^2+(\theta-\theta_i)^2\bigr),
\qquad 0<\varepsilon\ll 1,
\]
and glued by cutoffs into a global test function
\[
U_\varepsilon(r,\theta)
=
\log\!\Bigl[
\sum_{i=1}^N
\chi_i(r,\theta)\,
\exp\!\bigl(\phi_{\varepsilon,i}(r,\theta)+\ln \nu_i\bigr)
\Bigr].
\]
For its trace \(f_\varepsilon\), the moment constraints are exact, and the asymptotics are
\[
\int_{S^1} e^{f_\varepsilon}\,d\sigma
=
2\pi\bigl(\varepsilon^{-1}+O(1)\bigr),
\qquad
\int_{B^2} |\nabla U_\varepsilon|^2\,dx
=
4\pi(m+1)\log\frac1\varepsilon+O(1).
\]
Hence
\[
\frac{
\log\!\bigl[(2\pi)^{-1}\int_{S^1} e^{f_\varepsilon}\,d\sigma\bigr]
}{
\int_{B^2} |\nabla U_\varepsilon|^2\,dx
}
\longrightarrow
\frac1{4\pi(m+1)},
\]
so no smaller constant can hold [2107.08647].

## 5. Weighted Milin inequalities for logarithmic coefficients

The phrase “constrained Lebedev–Milin inequality” also appears in univalent function theory. Let
\[
f(z)=z+a_2 z^2+a_3 z^3+\cdots
\]
be analytic on the unit disk, and define the logarithmic coefficients by
\[
\log \frac{f(z)}{z}=2\sum_{n=1}^\infty \gamma_n z^n.
\]
For \(f\) univalent, de Branges’ theorem yields, for every integer \(N\ge 1\),
\[
\sum_{k=1}^N k(N-k+1)|\gamma_k|^2
\le
\sum_{k=1}^N \frac{N-k+1}{k},
\]
with equality for all \(N\) simultaneously if and only if \(f\) is the Koebe function \(K(z)=z/(1-z)^2\) or one of its rotations [1903.09974].

A general weighted version is obtained by replacing the coefficient \(N-k+1\) with a convex weight sequence. If \((p_n)_{n\ge 1}\) is a convex sequence of nonnegative real numbers satisfying
\[
p_1>0,\qquad \sum_{n=1}^\infty \frac{p_n}{n}<+\infty,
\]
then for every \(f\in \mathcal S\),
\[
\sum_{n=1}^\infty n\,p_n\,|\gamma_n|^2
\le
\sum_{n=1}^\infty \frac{p_n}{n}.
\]
Equality holds if and only if \(f(z)=z/(1-e^{i\theta}z)^2\) for some \(\theta\in \mathbb R\) [1903.09974].

The proof rewrites the convexity condition through
\[
\lambda_n=p_n-2p_{n+1}+p_{n+2}\ge 0,
\]
together with
\[
p_k=\sum_{n=k}^\infty \lambda_n (n-k+1),
\]
and then sums the classical Milin inequalities against \(\lambda_N\). This is a summation-by-parts reduction from the weighted statement to de Branges’ finite inequalities [1903.09974].

Roth’s extension relaxes global convexity. If \(\lambda_n\ge 0\) for all \(n>N\), one studies a finite weighted inequality through de Branges’ ODE system
\[
\tau_k(t)-\tau_{k+1}(t)
=
-\frac{\tau_k'(t)}{k}
-
\frac{\tau_{k+1}'(t)}{k+1},
\qquad
\tau_{N+1}(t)\equiv 0,
\]
and positivity of certain Jacobi-polynomial combinations
\[
Q_k(x)=\sum_{j=0}^{N-k}\nu_{k,j} P_j^{(2k,0)}(x).
\]
This permits sharp nonconvex examples, including
\[
\sum_{n=1}^\infty \frac{n^2}{(n+1)^2}|\gamma_n|^2
\le
\sum_{n=1}^\infty \frac1{n(n+1)^2}
=
\frac{\pi^2}{6}-1,
\]
and, for \(\alpha^2=4/3\),
\[
\sum_{n=1}^\infty \frac{n^2}{n^2+4/3}|\gamma_n|^2
\le
\sum_{n=1}^\infty \frac1{n^2+4/3}
=
\frac{(2\pi/\sqrt3)\coth(2\pi/\sqrt3)-1}{8/3}
\approx 0.98727
\]
[1903.09974].

## 6. Higher-order analogues and applications

The constrained Lebedev–Milin framework extends beyond second-order boundary analysis. For a compact manifold with boundary \((X^{n+1},g)\), Case and Luo construct a sixth-order GJMS operator \(L_6\), associated conformally covariant boundary operators \(B_j^5\), and the energy
\[
E_6(u)=\int_X u\,L_6u+\sum_{j=0}^2 \oint_M [B_j^5(u)][B_{5-j}^5(u)].
\]
On compactifications of Poincaré–Einstein manifolds, the boundary operators realize fractional GJMS operators of orders one, three, and five. In critical dimension \(n=5\), the sharp Onofri–Lebedev–Milin inequality on \(S^5\),
\[
\int_{S^5} w\,P_5 w
\ge
\frac{128}{5}\,\mathrm{Vol}(S^5)\,
\ln\!\Bigl(\int_{S^5} e^{5(w-\bar w)}\Bigr),
\]
yields the six-dimensional Lebedev–Milin or Onofri-type trace inequality
\[
E_6(u)
\ge
3\,C_{5,\tfrac12}\|\psi\|_{L^2}^2
+
8\,C_{5,\tfrac32}\|\phi\|_{L^2}^2
+
\frac{128}{5}\,\mathrm{Vol}(S^5)\,
\ln\!\Bigl(\int_{S^5} e^{5(f-\bar f)}\Bigr),
\]
where \(f=B_0^5(u)\), \(\phi=B_1^5(u)\), and \(\psi=B_2^5(u)\). Equality holds precisely when \(L_6u=0\) and each boundary datum has the Möbius-invariant extremal form recorded in the source [1810.08027].

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
\[
\log\!\Bigl(\int_T e^\phi\Bigr)-\int_T \phi
\le
\frac{1}{4(n+1)\pi}\int_D |\nabla \Phi|^2,
\]
under vanishing of the first \(n\) nonzero Fourier modes of \(e^\phi\), 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 \(\frac1{n+1}\)-periodic interaction satisfying
\[
2W(n+1)=1,\qquad 2W(k)\le \frac{n+1}{k}\quad (k\ge 1),
\]
the uniform distribution remains the unique minimizer for \(K\le 1\), and the transition occurs exactly at \(K=1\) [2604.16288].

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 \(W(\theta)=-|\sin(2\pi\theta)|\), the phase transition is continuous at
\[
K_c=K_\#=\frac{3\pi}{4}.
\]
For the noisy transformer model \(W_\beta(\theta)=(e^{\beta\cos(2\pi\theta)}-1)/\beta\), a sharp threshold \(\beta_*\) is identified such that \(K_c(\beta)=K_\#(\beta)\) and the phase transition is continuous for \(\beta\le \beta_*\), while \(K_c(\beta)<K_\#(\beta)\) and the phase transition is discontinuous for \(\beta>\beta_*\). An analogous sharp dichotomy is obtained for the noisy Hegselmann–Krause model \(W_R(\theta)=(R-2\pi|\theta|)_+^2\) [2604.16288].

Across these formulations, the common mechanism is the removal or controlled weighting of low modes: Fourier moments of the exponential measure on \(S^1\), center-of-mass modes on \(S^2\), 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 [1909.00431] [2109.13385] [1903.09974].

Source: https://www.emergentmind.com/topics/constrained-lebedev-milin-inequality