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Let n ≥ 1 be a fixed integer and R be a ring with involution ' ∗ '. For any two elements x and y in R, the n-skew Lie product and n-skew Jordan product are given by ▿ [ x , y ] n = ▿ [ x , ▿ [ x , y ] n - 1 ] with ▿ [ x , y ] 0 = y , ▿ [ x , y ] 1 = ▿ [ x , y ] = x y - y x ∗ , ▿ [ x , y ] 2 = x 2 y - 2 x y x ∗ + y (x ∗) 2 and x ⋄ n y = x ⋄ (x ⋄ n - 1 y) with x ⋄ 0 y = y , x ⋄ 1 y = x ⋄ y = x y + y x ∗ , x ⋄ 2 y = x 2 y + 2 x y x ∗ + y (x ∗) 2 . The purpose of this paper is to characterize Jordan left ∗ -centralizers satisfying certain functional identities involving skew Lie product and skew Jordan products. In particular, it is proved that if R is a 2-torsion free prime ring with involution of the second kind admits a non-zero Jordan left ∗ -centralizer T such that ▿ [ x , T (x) ] n ∈ Z (R) (n = 1 , 2) for all x ∈ R , then T (x) = λ x ∗ for all x ∈ R , where λ ∈ C , the extended centroid of R. We also characterize Jordan left ∗ -centralizers of prime rings with involution via skew Jordan product. [ABSTRACT FROM AUTHOR]

In this paper, we study some important means of Jordan's totient function, especially, we obtain asymptotic formula for geometric mean and harmonic mean. We also study alternating sums of Jordan's totient function and Carleman's inequality for this function. [ABSTRACT FROM AUTHOR]