---

Let $p=z^4+az^2+bz+c$ satisfy the claim. Its three critical points are simple, since it has three distinct critical values. Polynomial division gives

$$\begin{aligned} V_p(z)&=\frac{z^5}{4}+\frac{3a z^3}{8}+\frac{7b z^2}{16} +\left(\frac{a^2}{16}+\frac c2\right)z+\frac{3ab}{16},\\ Q_p(z)&=V_p'(z)=\frac54z^4+\frac{9a}{8}z^2+\frac{7b}{8}z +\frac{a^2}{16}+\frac c2,\\ R_p(z)&=p(z)^2-p'(z)V_p(z)=r_2z^2+r_1z+r_0,\\ r_2&=ac-\frac{a^3}{8}+\frac{9b^2}{16},\qquad r_1=\frac32bc-\frac{7a^2b}{16},\qquad r_0=c^2-\frac{3ab^2}{16}. \end{aligned} \tag{1}$$

A local family with prescribed critical values

Solve the holomorphic coefficient differential equation

$$\partial_\tau q_\tau=R_{q_\tau},\qquad q_0=p. \tag{2}$$

Its polynomial right-hand side gives a unique holomorphic solution for $\tau$ in a neighborhood of zero, consisting of centered monic quartics. The critical points remain simple there. At a moving critical point $c_j(\tau)$ its value $v_j(\tau)=q_\tau(c_j(\tau))$ satisfies

$$v_j'(\tau)=R_{q_\tau}(c_j(\tau))=v_j(\tau)^2, \qquad v_j(\tau)=\frac{v_j}{1-\tau v_j}. \tag{3}$$

Shrink the parameter neighborhood to lie in $|\tau|<1$ and define

$$p_\tau(z)=(1-|\tau|^2) q_\tau\bigl((1-|\tau|^2)^{-1/4}z\bigr)-\bar\tau. \tag{4}$$

This is a real analytic centered monic family. Its critical values are

$$\frac{v_j-\bar\tau}{1-\tau v_j}, \tag{5}$$

which lie on the unit circle and stay distinct. No global continuation or theorem about the critical-value map is needed here.

Differentiating the length

For $\zeta$ on the unit circle and different from the three $v_j$, each solution $w=w(\tau,\zeta)$ of

$$q_\tau(w)=\frac{\zeta}{1-\tau\zeta}$$

is locally holomorphic in $\tau$. Differentiating and using (2) gives

$$\partial_\tau w=V_{q_\tau}(w). \tag{6}$$

On each of the open arcs between consecutive $v_j$ choose the four inverse branches and let $h=\partial_\theta w$, where $\zeta=e^{i\theta}$. This tangent derivative is nonzero, and (6) implies

$$\partial_\tau\log h=Q_{q_\tau}(w),\qquad \partial_\tau^2\log h\big|_{\tau=0}=W_p(w), \tag{7}$$

where, by differentiating (1) in the coefficient direction $R_p$,

$$W_p=V_pQ_p'+\frac{9r_2}{8}z^2+\frac{7r_1}{8}z +\frac{ar_2}{8}+\frac{r_0}{2}. \tag{8}$$

The singular endpoints cause no loss of differentiability. Near a simple critical point the inverse branches have a convergent square-root expansion. The parameterization in (3) moves the endpoint and the target by the same fractional linear map, so that the singular factor is $|\theta-\theta_j|^{-1/2}$ at a fixed value of $\theta_j$. The expansion coefficients are holomorphic in $\tau$ and bounded on a smaller closed parameter disc, since the critical points remain simple and their values remain distinct. Consequently $|h|$ has a common integrable majorant. The factors $Q_{q_\tau}(w)$ and their first parameter derivatives are bounded there as well, since all the roots in question stay in a compact set. Thus the first and second parameter derivatives of $|h|$ have a common integrable majorant. Differentiation under the sum and integral below is justified.

The fractional linear image of the unit circle is the circle with center $\bar\tau/(1-|\tau|^2)$ and radius $(1-|\tau|^2)^{-1}$. The length of its inverse image under $q_\tau$ is

$$L(\tau)=\sum_{\text{arcs and branches}}\int |h(\tau,\theta)|\,d\theta, \qquad \ell(p_\tau)=(1-|\tau|^2)^{1/4}L(\tau). \tag{9}$$

The sum counts every regular arc of the lemniscate once; the finitely many critical points have zero arclength. If $h$ is nonzero and holomorphic in $\tau$, (7) gives at zero

$$\begin{aligned} \partial_\tau |h|&=\tfrac12Q|h|,\\ \partial_\tau\partial_{\bar\tau}|h|&=\tfrac14|Q|^2|h|,\\ \partial_\tau^2|h|&=(\tfrac12W+\tfrac14Q^2)|h|. \end{aligned} \tag{10}$$

Divide the integrals by $L(0)=\ell(p)$ and write $\mathbb E$ for normalized arclength expectation on $E(p)$. With $m=\mathbb E Q$, equations (9)--(10), including the derivative of its real prefactor, give for $u=\log\ell(p_\tau)$

$$\begin{aligned} 2u_\tau&=m,\\ 4u_{\tau\bar\tau}&=\mathbb E|Q|^2-|m|^2-1,\\ 2u_{\tau\tau}&=\mathbb E W+\tfrac12(\mathbb E Q^2-m^2). \end{aligned} \tag{11}$$

All derivatives in (11) are evaluated at zero. In particular $\Delta_\tau u=\operatorname{Var}(Q)-1$ and $\Delta_\tau\ell(p_\tau)/\ell(p)=\mathbb E|Q|^2-1$ there.

The Hessian in angular gaps

Label the critical values in counterclockwise order as $v_1,v_2,v_3$. Choose local real arguments $\theta_j(\tau)$ of (5), and let $\omega_1=(\theta_2-\theta_1)/(2\pi)$ and $\omega_2=(\theta_3-\theta_2)/(2\pi)$; the last gap is $1-\omega_1-\omega_2$. Put

$$\delta_j=v_{j+1}-v_j,\qquad \epsilon_j=v_{j+1}^2-v_j^2,\qquad j=1,2.$$

The real Jacobian at zero is

$$J=\frac1\pi \begin{pmatrix} \operatorname{Im}\delta_1&\operatorname{Re}\delta_1\\ \operatorname{Im}\delta_2&\operatorname{Re}\delta_2 \end{pmatrix}. \tag{12}$$

It is invertible: three distinct points of a circle are not collinear. Hence the gaps are real local coordinates for this family. Let $g$ be the two-component gradient of $\log\ell$ in these coordinates at $p$. By (11), it is determined by

$$J^{\mathsf T}g=(\operatorname{Re}m,-\operatorname{Im}m)^{\mathsf T}. \tag{13}$$

Expanding the arguments in (5) gives

$$\theta_j(\tau)=\theta_j(0)+2\operatorname{Im}(\tau v_j) +\operatorname{Im}(\tau^2v_j^2)+O(|\tau|^3).$$

Thus each $\omega_j$ is harmonic in $\tau$ and its real Hessian at zero is

$$H_j=\frac1\pi \begin{pmatrix} \operatorname{Im}\epsilon_j&\operatorname{Re}\epsilon_j\\ \operatorname{Re}\epsilon_j&-\operatorname{Im}\epsilon_j \end{pmatrix}. \tag{14}$$

The chain rule says that the Hessian in the gaps is $J^{-\mathsf T}(D^2_\tau u-g_1H_1-g_2H_2)J^{-1}$. The middle symmetric matrix has trace $\operatorname{Var}(Q)-1$ and eigenvalues

$$\frac{\operatorname{Var}(Q)-1}{2}\ \mathbin{\pm}\ |B|, \quad B=\mathbb E W+\frac{\mathbb E Q^2-m^2}{2} +\frac{i}{\pi}\sum_{j=1}^2g_j\epsilon_j. \tag{15}$$

Congruence by an invertible real matrix preserves positive semidefiniteness. Therefore convexity of $\log\ell$ in the gaps at $p$ is equivalent to the exact inequality

$$\boxed{\operatorname{Var}(Q)-1\ge2|B|.} \tag{16}$$

Equation (16) is a criterion, not an established inequality for every quartic. The weaker trace inequality and the stationary-point test following (11) are likewise unresolved in the linked research sketch.