§6.8 Inequalities

Notes:: See Gautschi (1959b) for (6.8.1), and Luke (1969b, p. 201) for (6.8.2) and (6.8.3).
Keywords:: exponential integrals, inequalities
Permalink:: http://dlmf.nist.gov/6.8

In this section $x>0$ .

6.8.1 $\frac{1}{2}\mathop{\ln\/}\nolimits\!\left(1+\frac{2}{x}\right)<e^{x}\mathop{E_{1}\/}\nolimits\!\left(x\right)<\mathop{\ln\/}\nolimits\!\left(1+\frac{1}{x}\right),$

Symbols:: $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 5.1.20
Referenced by:: §6.8
Permalink:: http://dlmf.nist.gov/6.8.E1
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6.8.2 $\frac{x}{x+1}<xe^{x}\mathop{E_{1}\/}\nolimits\!\left(x\right)<\frac{x+1}{x+2},$

Symbols:: $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral and $x$ : real variable
Referenced by:: §6.8
Permalink:: http://dlmf.nist.gov/6.8.E2
Encodings:: TeX, pMML, png

6.8.3 $\frac{x(x+3)}{x^{2}+4x+2}<xe^{x}\mathop{E_{1}\/}\nolimits\!\left(x\right)<\frac{x^{2}+5x+2}{x^{2}+6x+6}.$

Symbols:: $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral and $x$ : real variable
Referenced by:: §6.8
Permalink:: http://dlmf.nist.gov/6.8.E3
Encodings:: TeX, pMML, png