# How is it a convex function?

In one of the research paper it is mentioned that the following is a convex function $$f(v_1,\cdots v_N)=\frac{a_1v_1+\cdots +a_Nv_N}{c+\sum_{i=1}^Nd_iv_i}$$ where $$a_i,d_i,c$$ are some real constants. I do not know how it is a convex function. I have read in Convex Optimization book that such functions are quasi-convex. But is this function also convex?

• Not in this generality; say, for example, $$f(v)=\frac{v}{1+v}$$ is certainly not convex, but it has the form you gave, with $N=1, a_1=1, c=1, d_1=1$. – Giuseppe Negro Mar 5 at 9:59
• @GiuseppeNegro thank you for your comment. But we know that its quasiconvex function. So my next question is "Can we apply the KKT conditions to the problems with quasiconvex functions?" Thanks in advance. – Frank Moses Mar 5 at 10:06
• That's another question, you might want to ask it separately. – Giuseppe Negro Mar 5 at 10:08
• @GiuseppeNegro ok I have asked a new question. Please comment on that one too. Thanks in advance. The link the question is as follows math.stackexchange.com/questions/3136023/… – Frank Moses Mar 5 at 10:14
• A linear fractional transformation is pseudoconvex (in fact, pseudolinear), which is a stronger version of quasiconvexity. – A.Γ. Mar 5 at 22:29