A Continuous Description of Different Means with Application To Mixing Rules

Consider two positive nonzero numbers <i>x</i> ₁ and <i>x</i> ₂. In many cases, the arithmetic (AM), geometric (GM), or harmonic mean (HM) is used as an appropriate mean <i>x</i> ₁₂ of <i>x</i> ₁ and <i>x</i> ₂: AM = (<i>x</i> ₁ + <i>x</i> ₂)/2, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>GM</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow> <mml:mrow><mml:mn>1</mml:mn></mml:mrow> </mml:msub> <mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow> <mml:mrow><mml:mn>2</mml:mn></mml:mrow> </mml:msub> </mml:msqrt> </mml:math> , and HM = GM²/AM. However, sometimes it is not clear from the outset which mean value should be selected. This results in a discrete problem. Instead of this, here we discuss three one-parameter functions, which are able to describe a continuous connection between the mean values HM, GM, and AM, respectively. Two of these functions are known as the Lehmer mean and Hölder mean, whereby especially the Hölder mean is suitable for generating a uniform one-parameter description of various mixing rules as used in quantum chemical force field calculations and thermophysical properties calculations.