Sharp Beckner's inequalities for axially symmetric functions on S6 and S8

We prove that for $N=6$ and 8, axially symmetric solutions to the Q-curvature type problem$\alpha P_{N}u+(N-1)!(1-\frac{e^{Nu}}{{\int }_{S^N}{e}^{Nu}})=0\text{on}{S}^N$ must be constants, provided that $\frac{1}{2}\le \alpha <1$ and the center of mass of u is at the origin. We also show that this result is sharp. This result closes the gap of the related results in [21], which proved a similar uniqueness result for $\alpha \ge 0.6168$ when $N=6$ and $\alpha \ge 0.8261$ when $N=8$. As a consequence, we attain the best constant of sharp Beckner's inequality for axially symmetric functions on $S^6$ and $S^8$ whose center of mass is at the origin and answer the generalized Chang-Yang conjecture positively in the axially symmetric case when $N=6$ and $N=8$. The improvement is based on two types of new estimates. One is the refined estimate of the semi-norm ${\lfloor G\rfloor }^2$ using a new way of integration by parts. The other is a family of refined pointwise estimates (Lemma 3.7, Lemma 3.9) on Gegenbauer coefficients, which is established by the decaying estimates and cancellation property of Gegenbauer polynomials (Lemma 3.10, Proposition 3.11, Corollary 3.12). In particular, we use a three-fold line function when $N=8$ to further enhance the estimates of Gegenbauer polynomials.