Abstract In this article, we study the logarithmic Mahler measure of one-parameter family $$Q_\alpha=y^2+(x^2-\alpha x)y+x,$$ denoted by $\mathrm{m}(Q_\alpha)$. The zero loci Qα generically define elliptic curves Eα, which are 3-isogenous to Hessian curves. We particularly interested in case $\alpha\in (-1,3)$, has not been considered literature due certain subtleties. For α interval, establish...

We show that the \(\alpha\)-fractional bilinear indicator/cube testing constant \(\mathcal{BICT}_{T^{\alpha }}\left( \sigma ,\omega \right) \equiv \sup_{Q\in \mathcal{P}^{n}}\sup_{E,F\subset Q}\frac{1}{\sqrt{\left\vert Q\right\vert_{\sigma }\left\vert Q\right\vert _{\omega }}}\left\vert \int_{F}T_{\sigma}^{\alpha }\left( \mathbf{1}_{E}\right) \omega \right\vert ,\) defined for any singular in...