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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 $$\alpha_2=0.338\ldots,\;\beta_2={1\over \sqrt{2\pi}}=0.398\ldots$$.

##### MSC:
 33B15 Gamma, beta and polygamma functions 26D15 Inequalities for sums, series and integrals
##### Keywords:
gamma function; inequalities
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##### References:
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