Sharp trapezoid and mid-point type inequalities on closed balls in R3$\mathbb{R}^{3}$

This paper deals with some trapezoid and mid-point type inequalities on closed balls in $\mathbb{R}^{3}$R3. Three kinds of functions are considered: convex, Lipschitz, and bounded functions. The spherical coordinates are used to obtain sharp inequalities. Also a reverse result is given for the right-hand side of Hermite–Hadamard’s inequality obtained on closed balls in $\mathbb{R}^{3}$R3.

The main purpose of this paper is estimating two bounds B 1 and B 2 such that 1 4 3 πR 3 and 1 4πR 2 σ (C, R) f (x, y, z) dσ -1 4 3 πR 3 Depending on the properties of the function f and the radius R, different values will be obtained for B 1 and B 2 . We call (2) a mid-point type inequality due to the following result obtained in [2] and interpretation of Fig. 1.
According to (4), we have an estimate for the difference between the area under the graph of f , i.e., b a f (x) dx, and the area of rectangle abcd, i.e., (ba)f ( a+b 2 ) (see Fig. 1). Also we call (3) a trapezoid type inequality due to the following result and Fig. 2.
Before presenting our main results, here we obtain a new representation of (1) and also give a reverse type theorem.
If we consider a convex function f :B(C, R) → R and the change of coordinates whereD((a, b), R) is a closed disk centered at the point (a, b) having radius R > 0, then we obtain Choosing z = R 2x 2y 2 in the latter integrals, the fact that 1 + ( ∂z ∂x ) 2 + ( ∂z ∂y ) 2 = R √ R 2 -x 2 -y 2 = R z , and using the surface integral formula for σ (C, R) imply that Inequality (8) gives another representation for (1). In a special case for a convex function f : Now for a reverse type result, consider a continuous function f defined on a convex subset V ⊂ R 3 such that (8) holds for all closed balls included in V. Then f is convex on V because otherwise there would exist X , Y ∈ V and λ ∈ (0, 1) such that Since f is continuous on V, we can find R > 0 and a point Z = (ā,b,c) in a convex combination of X and Y such that (9) holds on the whole ofB((ā,b,c), R) ⊂ V. So by the change of coordinates (6) and structure presented in (7) forD((ā,b), R) andB((ā,b,c), R), we obtain that which is a contradiction and this proves the convexity of f on V.
In the following sections we consider convex, Lipschitz, and bounded functions to obtain some trapezoid and mid-point type inequalities on a closed ball. We use the spherical coordinates in calculating the integrals.

Convex functions
In this section we obtain trapezoid and mid-point type inequalities for the case that the partial derivative absolute values of a considered function with respect to the radius in spherical coordinates is convex. We need the following lemma. f (x, y, z) dV and Proof Consider the spherical transformation x(ρ, ϕ, θ ) = a + ρ cos θ sin ϕ, It is obvious that the Jacobian of this transformation is J = ρ 2 sin ϕ. So we have (10). For (11), consider the curve η : It is clear that η([0, π] × [0, 2π]) = σ (C, R) and then by integrating with respect to the surface (arc length) we get This proves (11).
The following is a sharp trapezoid type inequality related to (1), where we consider a function with convex partial derivative (with respect to the radius ρ) absolute values defined onB(C, R).
To show the sharpness of (12), consider the function f : With some calculations we obtain that and On the other hand, since | ∂f ∂ρ | = 1, From (16) and (17) we have the sharpness of (12).
Now we obtain the midpoint type inequality related to (1), where the partial derivative absolute value of considered function defined onB(C, R) is convex.
Remark 2.5 In the proof of Theorem 2.3, we can find the following inequality: Although (18) is not sharp, if we consider f (x, y, z) = x 2 + y 2 + z 2 for x, y, z ∈B(C, R), we will find that inequality (21) is sharp.
Remark 2.6 If we drop out the convexity condition of | ∂f ∂ρ | in Theorems 2.2, 2.3, and consider the condition instead of that, then we get the following Ostrowski type inequalities (see [19,20]) on a closed ball:

Lipschitz functions
In this section we consider Lipschitz functions with respect to the Euclidian norm to obtain some trapezoid and mid-point type inequalities onB(C, R).
If f :B(C, R) is Lipschitz with respect to the Euclidian norm with the constant L > 0, then for any x = (a + ρ 1 cos θ 1 sin ϕ 1 , b + ρ 1 sin θ 1 sin ϕ 1 , c + ρ 1 cos ϕ 1 ) and y = (a + ρ 2 cos θ 2 sin ϕ 2 , b + ρ 2 sin θ 2 sin ϕ 2 , c + ρ 2 cos ϕ 2 ), with some calculations we obtain that Also it is obvious that if f : V ⊆ R 3 → R is Lipschitz with a constant L > 0 on V, then it is continuous and so integrable on V. We need the following result.
The following trapezoid type inequality related to (1) for L-Lipschitz functions on B(C, R) holds.
Now by replacing (10) and (11) in (23) and then dividing the result by 4 3 πR 3 , we deduce the desired result.
It is not hard to see that f (a + ρ cos θ sin ϕ, b + ρ sin θ cos ϕ, c + ρ cos ϕ) ≥ 0 for all 0 ≤ ρ ≤ R, 0 ≤ ϕ ≤ π , and 0 ≤ θ ≤ 2π . Also for the case ρ = R, we have f (a + R cos θ sin ϕ, b + R sin θ cos ϕ, c + R cos ϕ) = 0. So we have For L-Lipschitz functions we can obtain a mid-point type inequality as follows: So we obtain that which implies the desired result.
Now consider the function f :B(C, R) → R defined by f (a + ρ cos θ sin ϕ, b + ρ sin θ cos ϕ, c + ρ cos ϕ) = Lρ, It is obvious that f (C) = 0. By a similar method used in the proof of Theorem 3.3, the function f is L-Lipschitz. So we have showing that inequality (24) is sharp.

Remark 3.5 Consider an open set
For convex function f defined on V, from Theorem D of Sect. 41 in [21] we have that f is L-Lipschitz onB(C, R) and so from inequalities (22) and (24), along with inequality (1), we get the following results: In the following, as an example we obtain a Lipschitz constant L for a real-valued function defined on a closed ball in R 3 .
for t ∈ [0, 1]. Now using the fundamental theorem of calculus, we obtain that On the other hand, from the chain rule for differentiation, we get where ∇f is the gradient vector of f . So using the Euclidean norm · , we obtain This shows that L = sup u∈B(C,R) ∇f (u) (if it exists) is a Lipschitz constant for f . Now for any w = (x, y, z) ∈B(C, R), we have and then So we can choose L = sup u∈B(C,R) ∇f (u) = nR n-1 as a Lipschitz constant for f onB(C, R).
Remark 3.9 In Theorems 3.3 and 3.4, if we consider that ∂f ∂ρ :B(C, R) → R is L-Lipschitz and f :B(C, R) → R is integrable, then by (13) and (19) we can obtain (the details are omitted)

Bounded functions
In the last section we investigate trapezoid and mid-point type inequalities where considered functions are bounded.
where L B and U B are lower and upper bounds of ∂f ∂ρ onB(C, R), respectively.
Proof Consider U B and L B as the upper and lower bounds of an arbitrary function g defined on a set V ⊂ R 3 , respectively. Then for all x, y, z ∈ V, we have which implies that for all x, y, z ∈ V. On the other hand, from (13)  Finally, by the use of the triangle inequality and dividing the result by 4πR 3 , we obtain inequality (26). If ∂f ∂ρ is bounded onB(C, R), then where L B and U B are lower and upper bounds of ∂f ∂ρ onB(C, R), respectively.
Proof Consider L B and U B as the upper and lower bounds of ∂f ∂ρ . By (19), the following relations hold: