A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

and Applied Analysis 3 called cofinite if the recession function f0 of f satisfies f0 y ∞, for all y / 0 see 15, page 116 . Then f is cofinite if and only if dom f∗ R by means of 15, Corollary 13.3.1 . The terminology “cofinite” is renewed as “coercive” in 16, 3.26 Theorem . Now we take a look at Atteia and Raı̈ssouli 11, Proposition 4.4 with a refined proof. Proposition 2.1 See Atteia and Raı̈ssouli 11, Proposition 4.4 . Let dom f ∩ dom g / ∅. If either f or g is cofinite, then all βn f, g and β∗ n f, g belong to Γ and βn f, g is cofinite for all n ≥ 0. Hence the geometric mean f # g due to Atteia and Raı̈ssouli [11], that is, the limit f # g lim n→∞ βn ( f, g ) , 2.3 is well defined and proper convex on dom β0 f, g . In particular, it belongs to Γ under the assumption that either dom β0 f, g dom β∗ 0 f, g or dom β0 f, g is closed. Moreover, f # g f ∗ # g∗ ∗ under the condition dom β0 f, g dom β∗ 0 f, g . Proof. Without loss of generality, we may assume that g is cofinite. Clearly, β0 f, g 1/2 f g ∈ Γ since dom β0 f, g dom f ∩ dom g / ∅. In addition, β0 f, g is still cofinite by 15, Theorem 9.3 . Then β∗ 0 f, g 1/2 f ∗ g∗ ∗ 1/2 f g ∈ Γ by virtue of 15, Corollary 9.2.2 . Thus dom β∗ 0 f, g 1/2 dom f dom g ⊇ dom β0 f, g . By induction, assume that βn ( f, g ) , β∗ n ( f, g ) ∈ Γ, βn(f, g) is cofinite, dom βn(f, g) ⊆ dom β∗ n(f, g). 2.4 Then dom βn 1 f, g dom βn f, g ∩ dom β∗ n f, g dom βn f, g , so βn 1 f, g ∈ Γ. Moreover, βn 1 f, g is cofinite because βn f, g is cofinite. It is readily checked that β∗ n 1 ( f, g ) ( βn 1 ( f∗, g∗ ))∗ (1 2 ( βn ( f, g ))∗ 1 2 ( β∗ n ( f, g ))∗)∗ . 2.5 Hence β∗ n 1 f, g 1/2 βn f, g β∗ n f, g ∈ Γ. In this case, dom β∗ n 1 f, g 1/2 dom βn f, g dom β∗ n f, g ⊇ dom βn f, g dom βn 1 f, g . Thus we obtain that ∀n, dom βn ( f, g ) dom f ∩ dom g dom β0 ( f, g ) , ∀n, dom β∗ n ( f, g ) ⊇ dom βn(f, g) dom β0(f, g). 2.6 According to Atteia and Raı̈ssouli 11, Proposition 4.4 , we have βn 1 ( f, g ) − β∗ n 1(f, g) ≤ 1 2 ( βn ( f, g ) − β∗ n(f, g)), ∀n ≥ 0; β∗ 0 ( f, g ) ≤ · ≤ β∗ n(f, g) ≤ β∗ n 1(f, g) ≤ · ≤ βn 1(f, g) ≤ βn(f, g) ≤ · ≤ β0(f, g). 2.7 Hence the geometric mean f # g is well defined and belongs to Γ under the given hypothesis. If dom β0 f, g is closed, we define an increasing sequence γn f, g ∈ Γ by γn ( f, g ) β∗ n ( f, g ) δC, 2.8 4 Abstract and Applied Analysis where δC denotes the indicator function of the closed convex set C dom β0 f, g . Obviously, f # g is the common limit of βn f, g and γn f, g , hence, belongs to Γ. For the equality f # g f∗ # g∗ ∗, we have ( f∗ # g∗ )∗ x sup y∈Rn [〈 y, x 〉 − (f∗ # g∗)(y)] sup y∈Rn [〈 y, x 〉 − lim n→∞ βn ( f∗, g∗ )( y )] sup y∈Rn [〈 y, x 〉 − lim n→∞ β∗ n ( f∗, g∗ )( y )] sup y∈Rn [〈 y, x 〉 − lim n→∞ ( βn ( f, g ))∗( y )] ≤ sup y∈Rn [〈 y, x 〉 − (βn(f, g))∗(y)], ∀n ( βn ( f, g ))∗∗ x βn(f, g) x , ∀n. 2.9 Hence ( f∗ # g∗ )∗ x ≤ lim n→∞ βn ( f, g ) x ( f # g ) x . 2.10 On the other hand, ( f∗ # g∗ )∗ x sup y∈Rn [〈 y, x 〉 − (f∗ # g∗)(y)] sup y∈Rn [〈 y, x 〉 − lim n→∞ βn ( f∗, g∗ )( y )] ≥ sup y∈Rn [〈 y, x 〉 − βn(f∗, g∗)(y)], ∀n ( βn ( f∗, g∗ ))∗ x β∗ n(f, g) x , ∀n. 2.11 Thus ( f∗ # g∗ )∗ x ≥ lim n→∞ β∗ n ( f, g ) x ( f # g ) x . 2.12