I. Knop J, Teasdale TW, Schulsinger F, Goodwin DW. A prospective study of young men at high risk for alcoholism: School behaviour and achievement. J Stud Alc 1985;46:273-278. II. Schulsinger F, Knop J, Goodwin DW, Teasdale TW, Mikkelsen U. A prospective study of young men at high risk for alcoholism: Social and psychological characteristics. Arch Gen Psychiatry 1986,43:755-760. III. Knop J, Goo...

During a 6-month period, 5,375 clinical specimens were cultured on Middlebrook-Cohn 7H10 medium, on Lowenstein-Jensen medium, and in Middlebrook 7H12 medium containing [14C]palmitic acid. More mycobacteria were recovered when all three media were used than when either the conventional method with 7H10 agar and Lowenstein-Jensen slants or the radiometric method with 7H12 broth was used alone.

In this paper, we establish the conditional stability of the generalized Jensen functional equation f (ax+by) = ag(x)+bh(y) on various restricted domains such as inside balls, outside balls, and punctured spaces. In addition, we prove the orthogonal stability of this equation and study orthogonally generalized Jensen mappings on balls in inner product spaces.

In this paper, we establish the conditional Hyers-Ulam-Rassias stability of the generalized Jensen functional equation r f ( sx+ty r ) = s g(x) + t h(y) on various restricted domains such as inside balls, outside balls, and punctured spaces. In addition, we prove the orthogonal stability of this equation and study orthogonally generalized Jensen mappings on Balls in inner product spaces.

Symmetry is repetitive self-similarity. We proved the stability problem by replicating well-known Cauchy equation and Jensen into two variables. In this paper, we Hyers-Ulam of bi-additive functional f(x+y,z+w)=f(x,z)+f(y,w) bi-Jensen 4fx+y2,z+w2=f(x,z)+f(x,w)+f(y,z)+f(y,w).

For the classical Jensen inequality of convex functions, i.e., [Formula: see text] an equivalent condition is proved in the framework of the generalized Sugeno integral. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral are given.

Jensen–Steffensen type inequalities for P -convex functions and functions with nondecreasing increments are presented. The obtained results are used to prove a generalization of Čebyšev’s inequality and several variants of Hölder’s inequality with weights satisfying the conditions as in the Jensen–Steffensen inequality. A few well-known inequalities for quasi-arithmetic means are generalized.