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In the article, we present the greatest values α and λ , and the least values β and μ in [0,1/2] such that the double inequalities H[αa+(1−α)b,αb+(1−α)a] < TQ(a,b) < H[βa+(1−β)b,βb+(1−β)a], G[λa+(1−λ)b,λb+(1−λ)a] < TQ(a,b) < G[μa+(1−μ)b,μb+(1−μ)a] hold for all a,b > 0 with a = b , where H(a,b) = 2ab/(a+b) , G(a,b) = √ab and TQ(a,b) = 2 π ∫ π/2 0 a cos2 θ bsin 2 θ dθ are respectively the harmonic, geometric and Toader-Qi means of a and b .

Sharp bounds for the Toader-Qi mean in terms of harmonic and geometric means