Hi everyone. In a proof I'm working on, I have a ratio of two sums of functions in the following form:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\frac{f_1(x)+f_2(x)+...f_n(x)}{g_1(x)+g_2(x)+...+g_n(x)}[/itex]

I want to prove this ratio is monotonically increasing in [itex]x[/itex]. All of the functions [itex]f_i(x)[/itex] and [itex]g_i(x)[/itex] are positive and also (importantly) I know that for all [itex]i=1,2,...,n[/itex], the ratio [itex]f_i(x)/g_i(x)[/itex] is monotonically increasing in [itex]x[/itex], i.e. [itex]f_1(x)/g_1(x)[/itex] is increasing in [itex]x[/itex], [itex]f_2(x)/g_2(x)[/itex] is increasing in [itex]x[/itex], etc.

Is there a simple way to prove this without requiring further information about these functions? I've been stuck on it for a while. Does it have to be true that the ratio of the sums is increasing? If anyone can suggest a straightforward approach (or tell me if it's not possible without further information) I'd be very grateful, thanks!

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Proving monotonicity of a ratio of two sums

Loading...

Similar Threads - Proving monotonicity ratio | Date |
---|---|

I Limits of Monotonic Sequences | Mar 11, 2018 |

A How can I Prove the following Integral Inequality? | Jan 24, 2018 |

I Proving that square root of 2 exists | Sep 24, 2017 |

I Proving non homeomorphism between a closed interval & ##\mathbb{R}## | Jul 10, 2017 |

A Proving Reimann Hypothesis True With Generalized Sets | Feb 10, 2017 |

**Physics Forums - The Fusion of Science and Community**