Suppose that we have some zk>0, k={0,1,⋯,n}. I want to compare weighted averages of zk's when the weights are defined by binomial probabilities.

More specifically, for p and q, where p,q∈(0,1), and for some λ∈(0,1), let x=λp+(1−λ)q.

In this case, should the following be true for all λ∈(0,1)?

$$max{\textstyle \{}\sum _{k=0}^{n}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}{p}^{k}(1-p{)}^{n-k}{z}_{k},\sum _{k=0}^{n}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}{q}^{k}(1-q{)}^{n-k}{z}_{k}{\textstyle \}}\ge \sum _{k=0}^{n}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}{x}^{k}(1-x{)}^{n-k}{z}_{k}.$$

I could see that it's true for $n=2$ but I can't show that it still holds for any $n>2$.