An integral inequality concerning isotropic measures on the unit circle (Q1827111)

Let \(f\) be a positive, continuous function defined on the unit circle, \(f:\mathbb{T} \to (0, \infty)\). The authors consider the extreme value problem for the function \[ F_j(a,\alpha)=\frac{1}{2\pi}\int^{2\pi}_0 \frac{f(\theta)\,d\theta} {(a^2{\cos}^2(\theta+\alpha)+a^{-2}{\sin}^2(\theta+\alpha))^{j/2}} \] for any \(a>0\) and any \(\alpha, j \in \mathbb{R}\). This problem can be interpreted as the extreme value problem for the dual quermassintegral \(\widetilde{W}_j\), restricted on some special family of star-shaped bodies in \(\mathbb{R}^2\) parameterized by elements of the group SL\((2, \mathbb{R})\). Without loss of generality the original problem is reduced to finding necessary and sufficient conditions for \(F_j\) to attain its extreme value for \(a=1\) and \(\alpha =0\). The main result of the paper under review is Theorem 1.4, in which the authors specify some condition on \(f\) which guarantees that \(F_j\) attains its maximal value at the point \((a,\alpha)=(1,0)\) (for all \(j\) close to \(1\)). To prove Theorem 1.4 the authors use some techniques involving Fourier coefficients and estimates of hypergeometric functions.