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Inequalities involving \(\Gamma (x)\) and \(\Gamma (1/x)\). (English) Zbl 1091.33001
The author improves in this paper upon the result by W. Gautschi [SIAM J. Math. Anal. 5, 278–281 (1974; Zbl 0239.33002); SIAM J. Math. Anal. 5, 282–292 (1974; Zbl 0239.33003)]; \[ 1\leq {2\over {1\over \Gamma(x)}+{1\over \Gamma(1/x)}}\leq 2\quad (x>0). \] After the formulation and proof of 11 lemmas, the main result follows in the form \[ {1\over x+1/x}+{\alpha\over (x+1/x)^{1/2}}\leq {1\over \Gamma(x)}+{1\over\Gamma(1/x)}\leq {1\over x+1/x}+{\beta\over (x+1/x)^{1/2}}\quad (x>0), \] with best possible constants \(\alpha=0,\;\beta=3/\sqrt{2}\).
The paper concludes with two sections on several results concerning products/sums and differences/quotients of gamma functions at reciprocal arguments:
A. For all \(x\in (0,1]\): \[ {1\over x}+\alpha^{\ast}\left(x-{1\over x}\right)\leq {\Gamma(x)\Gamma(1/x)\over \Gamma(x+1/x)}\leq {1\over x}+\alpha^{\ast}\left(x-{1\over x}\right), \] with best constants \(\alpha^{\ast}=1/2,\;\beta^{\ast}=0\).
B. For all real \(x>0\): \[ \alpha_0\leq{\Gamma(x)+\Gamma(1/x)\over\Gamma(x+1/x)}\leq \beta_0, \] with best possible constants \(\alpha_0=0.770\ldots,\;\beta_0=2.097\ldots\)
C. For all \(x\in (0,1]\): \[ \alpha_1 x^{1-1/x}\leq \Gamma(x)-\Gamma(1/x)\leq \beta_1 x^{1-1/x}, \] with best possible constants \(\alpha_1=-0.040\ldots,\;\beta_1=0.420\ldots\)
D. For all \(x\in (0,1]\): \[ \alpha_2\,{(ex)^{1/x}\over x\sqrt{x}}\leq{\Gamma(x)\over\Gamma(1/x)}\leq\beta_2\,{(ex)^{1/x}\over x\sqrt{x}}, \] with best possible constants \(\alpha_2=0.338\ldots,\;\beta_2={1\over \sqrt{2\pi}}=0.398\ldots\).

33B15 Gamma, beta and polygamma functions
26D15 Inequalities for sums, series and integrals
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