Expert: Paul Klarreich - 9/22/2007

Question
Dear Paul. I am an adult with a general math question. I received a response that my question was not clear enough so here is a rephrased version, hopefully lucid enough. The logarithmic function y = a log(x) + b is a very popular example for a monotonically increasing function that has a concave-down shape. Another example is the power law function y = b x^a. It also generates monotonically increasing concave-down shape for 0 < a < 1 (for example the square root function is a = 0.5). My question is which other general functional forms (that is, in addition to the logarithmic and the power-law functions as defined above) produce a monotonically increasing concave-down shape? Let me be even more clear - all monotonically increasing sigmoid-shape functions are concave up (=convex) at relatively low values of x – I am looking for a function which is (for a given a and b) concave down for any range of positive x values.

Answer
Questioner:   Joel Jones
Category:  Advanced Math
Private:  No
 
Subject:  concave down functions

Question:  Dear Paul. I am an adult with a general math question. I received a response that my question was not clear enough so here is a rephrased version, hopefully lucid enough. The logarithmic function y = a log(x) + b is a very popular example for a monotonically increasing function that has a concave-down shape. Another example is the power law function y = b x^a. It also generates monotonically increasing concave-down shape for 0 < a < 1 (for example the square root function is a = 0.5). My question is which other general functional forms (that is, in addition to the logarithmic and the power-law functions as defined above) produce a monotonically increasing concave-down shape? Let me be even more clear - all monotonically increasing sigmoid-shape functions are concave up (=convex) at relatively low values of x. I am looking for a function which is (for a given a and b) concave down for any range of positive x values.

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Hi, Joel,

I am not familiar with any specific class of functions that is always concave down, aside from the ones you mention.  In fact, the logarithmic function you give is not defined for all values of x, either, nor is the  x^(fraction).  However, I cannot give a proof that no such function exists.
Sorry, but that's the best I can do.