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I'm trying to study the sign of an expression, having information on some of my parameters. I don't know how to code these conditions. For example, I want to know when this expression:

k [2(a'[x])^2 - a[x]a''[x]] / (a[x]^2) 

is positive given that k > 0, x > 0 and that my function a is positive and increasing for all x > 0.

So basically, since a''[x] is the only term of the equation that I don't know anything about, I want Mathematica to give me:

(* a''[x] < a'[x]^2/a[x] *)

I'm trying to study the sign of an expression, having information on some of my parameters. I don't know how to code these conditions. For example, I want to know when this expression:

k (2(a'[x])^2 - a[x]a''[x]) / (a[x]^2) 

is positive given that k > 0, x > 0 and that my function a is positive and increasing for all x > 0 (k is a constant, a is a function)

So basically, since a''[x] is the only term of the equation that I don't know anything about, I want Mathematica to give me:

(* a''[x] < a'[x]^2/a[x] *)

I'm trying to study the sign of an expression, having information on some of my parameters. I don't know how to code these conditions. For exemple, I want to know when this expression-

k [2(a'[x])^2 - a[x]a"[x]] / (a[x]^2)

-is positive, knowing that k>0, x>0 and that my function "a" is positive and increasing for all x>0.

So basically, since a"[x] is the only term of the equation that I don't know anything about, I want Mathematica to gives me:

Out= a"[x] < a'[x]^2/a[x]

Thank you so much for your help,

PS: Alexei Boulbitch thank you for your answer, I hope it's clearer this time. I clearly have to get better at formulating programming questions!

I'm trying to study the sign of an expression, having information on some of my parameters. I don't know how to code these conditions. For example, I want to know when this expression:

k [2(a'[x])^2 - a[x]a''[x]] / (a[x]^2) 

is positive given that k > 0, x > 0 and that my function a is positive and increasing for all x > 0.

So basically, since a''[x] is the only term of the equation that I don't know anything about, I want Mathematica to give me:

(* a''[x] < a'[x]^2/a[x] *)

I would like to study the sign of an expression including a function and its 1st and 2nd derivatives. My function is positive and strictly increasing but I don't want to have to specify its form. I can't find a way to make these 2 statements, so that the condition would be on the 2nd derivative. I tried "Assumptions", "ForAll[x, f[x]>0 && f'[x]>0]","/;{f[x]>0,f'[x]>0}","If"..but nothing works. I didn't succeed either on programming easier examples like: if f'[x]>0 and x<y, then f[x]<f[y] -> True or 2+a>2/;a>0 -> True Thank you very much for your help

I'm trying to study the sign of an expression, having information on some of my parameters. I don't know how to code these conditions. For exemple, I want to know when this expression-

k [2(a'[x])^2 - a[x]a"[x]] / (a[x]^2)

-is positive, knowing that k>0, x>0 and that my function "a" is positive and increasing for all x>0.

So basically, since a"[x] is the only term of the equation that I don't know anything about, I want Mathematica to gives me:

Out= a"[x] < a'[x]^2/a[x]

Thank you so much for your help,

PS: Alexei Boulbitch thank you for your answer, I hope it's clearer this time. I clearly have to get better at formulating programming questions!