Definition 4.3 A Young function Φ is said to be of upper type p (resp. lower type p) for some p ∈ [ 0, ∞ ), if there exists a positive constant C such that, for all t ∈ [ 1, ∞ ) (resp. t ∈ [ 0, 1 ] ) and s ∈ [ 0, ∞ ), Φ ( s t ) ≤ C t p Φ ( s ).
Clearly, varphi (x, t):= w(x Phi(t quad mbox{for all } xin{{mathbb {R}}^{n}} mbox{ and } tin[0, infty ) satisfies Assumption if w is a classical or an anisotropic (mathcal{A}_{infty}(A)) Muckenhoupt weight (see, for example, [21]) and Φ is of lower type (p^_{varphi}) for some (p^_{varphi}in 0, infty )) and of upper type (p^_{varphi}) with (p^_{varphi}in (0, infty )).
With the definitions of upper-type set relation, see Definition 2, and minmax-robust efficiency in mind we can see the close connection between minmax-robust efficiency and the upper-type set relation, since a solution x ∈ X to ( P ( U ) ) is minmax-robust efficient if there is no other feasible solution x ¯ ∈ X ∖ { x }, such that f U ( x ¯ ) ⪯ C u f U ( x ), where Y = R k and C = R ≧ k.
By U q we denote the family of all growth functions Φ of positive upper type q (q ⩾ 1 ), such that the function t ↦ Φ (t ) / t is nondecreasing on [ 0, ∞ ).
By Lemma 3.2, in the future, we always consider a Musielak-Orlicz function φ of uniformly lower type (p^_{varphi}) and of uniformly upper type (p^_{varphi}), and (varphi (x, cdot)) is continuous and strictly increasing for all (xin{{mathbb{R}}^{n}}).
On the other hand, since φ is of uniformly upper type (p^_{varphi}in (0, infty )), we get, for any (xin{{mathbb{R}}^{n}}), tilde{varphi} (x, t)= int^{t}_{0}frac{varphi (x, s)}{s},dsgeq C frac{varphi (x,t)}{t^{p^_{varphi}}} int^{t}_{0}frac{1}{s^{1-p^_{varphi}}},dsgeq Cvarphi (x,t), where C is a positive constant as in (2.4).
It is said that the function Φ is of positive upper type (respectively, negative upper type), if there are q > 0 (respectively, q < 0 ) and C > 0 such that Φ (st ) ⩽ Ct q Φ (s ) for every s > 0 and t ⩾ 1.
There is no "missed opportunity" for late reconstructions for patients with upper type of birth palsies even if they underwent early nerve reconstruction operations.
More precisely, for (pin -infty,infty )), a function Φ is said to be of upin -infty. lower) type p,inftyhere exists a positive constant C such that, function(sin[1,infty )) (resp. (sin[0,1])) and (tin (0,infty )), Phi(st)leq Cs^{p}Phi(t).
Hand amputation was the most common type of upper limb amputations and below-knee amputation was the most common type of lower limb amputations constituting 8.8% and 30.5% of total cases, respectively.
"I mean, you're supposed to use whatever happen to you as some type of upper, not a downer.
