Complex Convexity of Musielak-Orlicz Function Spaces Equipped with the p-Amemiya Norm

and Applied Analysis 3 find d > 0 such that μ{t ∈ T : k 0 |x(t)| + d < e(t)} > 0. Indeed, if μ{t ∈ T : k 0 |x(t)| + d < e(t)} = 0 for any d > 0, then we set d n = 1/n and define the subsets T n = {t ∈ T : k 0 |x (t)| + dn < e (t)} (16) for each n ∈ N. Notice that T 1 ⊆ T 2 ⊆ ⋅ ⋅ ⋅ ⊆ T n ⊆ ⋅ ⋅ ⋅ and {t ∈ T : k 0 |x(t)| ≤ e(t)} = ⋃ ∞ n=1 T n ; we can find that μ {t ∈ T : k 0 |x (t)| ≤ e (t)} = 0 (17) which is a contradiction. Let T 0 = {t ∈ T : k 0 |x(t)| + d < e(t)} and define y = (d/k 0 )χ T0 ; we have y ̸ = 0 and for any λ ∈ C with |λ| ≤ 1, 󵄩󵄩󵄩󵄩x + λy 󵄩󵄩󵄩󵄩Φ,p ≤ 1 k 0 {1 + I p Φ (k 0 (x + λy))} 1/p = 1 k 0 {1 + [I Φ (k 0 xχ T\T0 ) +I Φ (k 0 xχ T0 + λdχ T0 )] p } 1/p ≤ 1 k 0 {1 + [I Φ (k 0 xχ T\T0 ) +I Φ ((k 0 |x| + d) T0 )] p } 1/p = 1 k 0 {1 + [I Φ (k 0 xχ T\T0 )] p } 1/p


Introduction
Let (, ‖ ⋅ ‖) be a complex Banach space over the complex field C, let  be the complex number satisfying  2 = −1, and let () and () be the closed unit ball and the unit sphere of , respectively.In the sequel, N and R denote the set of natural numbers and the set of real numbers, respectively.
In the early 1980s, a huge number of papers in the area of the geometry of Banach spaces were directed to the complex geometry of complex Banach spaces.It is well known that the complex geometric properties of complex Banach spaces have applications in various branches, among others in Harmonic Analysis Theory, Operator Theory, Banach Algebras,  * -Algebras, Differential Equation Theory, Quantum Mechanics Theory, and Hydrodynamics Theory.It is also known that extreme points which are connected with strict convexity of the whole spaces are the most basic and important geometric points in geometric theory of Banach spaces (see [1][2][3][4][5][6]).
In [7], Thorp and Whitley first introduced the concepts of complex extreme point and complex strict convexity when they studied the conditions under which the Strong Maximum Modulus Theorem for analytic functions always holds in a complex Banach space.
A point  ∈ () is said to be a complex extreme point of () if for every nonzero  ∈  there holds sup ||≤1 ‖+‖ > 1.A complex Banach space  is said to be complex strictly convex if every element of () is a complex extreme point of ().
In [8], we further studied the notions of complex strongly extreme point and complex midpoint locally uniform convexity in general complex spaces.
A point  ∈ () is said to be a complex strongly extreme point of () if for every  > 0 we have Δ  (, ) > 0, where A complex Banach space  is said to be complex midpoint locally uniformly convex if every element of () is a complex strongly extreme point of ().
For  ∈   (), define the -Amemiya norm by the formula For convenience, from now on, we write

Main Results
We begin this section from the following useful lemmas.
Lemma 1 (see [9]).For any  > 0, there exists  ∈ (0, 1/2) then where It follows from the definition of -Amemiya norm that there is a sequence {  } satisfying For any  > 0, we have If  >  0 /, notice that which means the sequence {  } is bounded.Hence, without loss of generality, we assume that   →  0 as  → ∞.
We can also choose the monotonic increasing or decreasing subsequence of {  } that converges to the number  0 .Applying Levi Theorem and Lebesgue Dominated Convergence Theorem, we obtain which implies  0 ∈   ().