Lower bound for balanced sets

We deal with small size balanced subsets of Zp when p is prime. In a balanced set S, each element x∈S is a midpoint between two other elements from S, i.e. , y,z∈S∖{x}. We denote the minimum cardinality of a balanced set modulo p by α(p).We prove a new lower bound for α(p). Thus we demonstrate that for every prime p>2, α(p)≥log2p+1.41.