Differential subordinations involving arithmetic and geometric means

The paper concerns the differential subordination with the expression combined by arithmetic and geometric means:α[p(z)]δ+(1-α)p(z)+zp′(z)p(z)μ≺φ(z),(p(0)=φ(0)=1,|z|<1),where δ,μδ,μ and α   are real numbers such that δδ, μ∈〈1,2〉μ∈〈1,2〉, α∈〈0,1〉α∈〈0,1〉. For δ∈〈1,2〉δ∈〈1,2〉, μ∈〈0,1〉μ∈〈0,1〉, α∈〈0,1〉α∈〈0,1〉 we also study the differential subordinationα[p(z)]δ+(1-α)[p(z)]μp(z)+zp′(z)p(z)1-μ≺φ(z),(p(0)=φ(0)=1,|z|<1).Several applications of the studied subordination in the theory of analytic functions are given.