Some new generalized concepts in this point of view are quasiconvex , strongly convex , approximately convex , logarithmically convex
, midconvexfunctions , pseudoconvex , [phi]-convex , [lambda]-convex , h-convex , delta-convex , Schur convex [11-15], and others [16-19].
which implies that the function [[gamma].sup.(m).sub.q](x,z) is logarithmically convex
. Also, since [[gamma].sup.(0).sub.q](x, z) = [[gamma].sub.q](x, z), then it follows that [[gamma].sub.q](x, z) is also logarithmically convex
Srivastava, "Alternative proofs for monotonic and logarithmically convex
properties of one-parameter mean values," Applied Mathematics and Computation, vol.
First we recall some definitions and facts about exponentially convex and logarithmically convex
functions (see, e.g., [13,14] or ) which we need for our results.
(a) [GAMMA] is a logarithmically convex
The function f given by (15) is strictly logarithmically convex
on (0, [infinity]) for k [greater than or equal to] 1.
is strictly decreasing and strictly logarithmically convex
on (0, [infinity]).