New variants of Khintchine's inequality (Q5944986)

Let \(r_n(t)= \text{sign}(\sin 2^n\pi t)\), \(t\in [0,1]\), \(n= 1,2,\dots\), be the sequence of usual Rademacher functions on \([0,1]\) and let \(x= (x_1,\dots, x_n)\) be a vector in \(\mathbb{R}^n\). Denoting \[ S(x)= \Biggl\|\sum^n_{i= 1} x_ir_i\Biggr\|_{L_1}- \Biggl\|{1\over\sqrt n} \sum^n_{i=1} r_i\Biggr\|_{L_1}\|x\|_2, \] we obtain upper and lower bounds for \(S(X)\) in terms of \(\|x\|_1\), \(\|x\|_\infty\) and \(\|x\|_2\) with coefficients depending on \(n\). For example, the following variants of Khinchin's inequality are proved: \[ {1\over 2}(\|x\|_1- \sqrt n\|x\|_2)\leq S(x)\leq \Biggl(1-\Biggl\|{1\over\sqrt n} \sum^n_{i=1} r_i\Biggr\|_{L_1}\Biggr) \|x\|_\infty, \] with equality for \(x= (1,1,\dots, 1)\) and for \(x= (1,0,\dots, 0)\), respectively. The Schur convexity of certain attached functions and direct estimates in terms of the Haagerup type of functions are used. Relations with known results are discussed and inequalities of independent interest are also proved.