ON BOUNDS OF THE SINE AND COSINE ALONG A CIRCLE ON THE COMPLEX PLANE

In the paper, the author finds bounds of the sine and cosine along a circle on the complex plane in terms of two double inequalities for the norms of the sine and cosine along a circle on the complex plane.


Motivations
In the theory of complex functions, the sine and cosine functions sin z and cos z on the complex plane C are defined by sin z = e iz − e −iz 2i and cos z = e iz + e −iz 2 respectively, where z = x + iy, x, y ∈ R, and i = √ −1 is the imaginary unit. They have the least positive periodicity 2π, that is, sin(z + 2kπ) = sin z and cos(z + 2kπ) = cos z for k ∈ Z.
When restricting z = x ∈ R, the sine and cosine functions sin z and cos z become sin x and cos x and satisfy 0 ≤ | sin x| ≤ 1 and 0 ≤ | cos x| ≤ 1.
When restricting z = iy for y ∈ R, the sine and cosine functions sin z and cos z reduce to sin(iy) = e −y − e y 2i = i sinh y → ±i∞ and cos(iy) = e −y + e y 2 = cosh y → +∞ as y → ±∞. These imply that the sine and cosine are bounded on the real x-axis, but unbounded on the imaginary y-axis.
In [6], a criterion to justify a holomorphic function was discussed.
In [5], the author discussed and computed bounds of the sine and cosine functions sin z and cos z along straight lines on the complex plane C. The main results in the paper [5] can be recited as follows.
(1) The complex functions sin z and cos z are bounded along straight lines parallel to the real x-axis on the complex plane C: (a) along the horizontal straight line y = α on the complex plane C, and where α ∈ R is a constant and x ∈ R; (b) the equalities in the left hand side of (1) and in the right hand side of (2) hold if and only if x = kπ for k ∈ Z; (c) the equalities in the right hand side of (1) and in the left hand side of (2) hold if and only if x = kπ + π 2 for k ∈ Z. where γ ∈ R is a constant; In this paper, we find bounds of the sine and cosine functions sin z and cos z along a circle centered at the origin z = 0 of radius r on the complex plane C in terms of double inequalities for their norms.

A double inequality for the norm of sine along a circle
In this section, we find a double inequality for the sine function.
Theorem 2.1. Let r > 0 be a constant and let C(0, r) : z = re iθ for θ ∈ [0, 2π) denote a circle centered at the origin z = 0 of radius r. Then The left equality is valid if and only if θ = 0, π while the right equality is valid if and only if θ = π 2 , 3π 2 .
Proof. The circle C(0, r) can be represented by It is not difficult to see that, for fixed r > 0, sin re iθ = | sin r| for θ = 0, π, sin re iθ = sinh r for θ = π 2 , 3π 2 , and sin re iθ has a least positive periodicity π with respect to the argument θ.
In Figure 1, we plot the 3D graph of sin re iθ for r ∈ [0, 5] and θ ∈ [0, 2π). In Figure 2, we plot the graph of sin πe iθ for θ ∈ [0, 2π). These two figures are helpful for analyzing and understanding the behaviour of the sine function sin z along the circle C(0, r) centered at the origin z = 0 of radius r.
Considering the odevity of sinh t and sin t, we see that two inequalities in (4) and (5) are equivalent to sinh t t > 1 and sin t t < 1 for t ∈ (0, ∞). The first inequality in (6) follows from cosh x > 1 for x = 0 and the Lazarević inequality in [2, p. 270, 3.6.9]. When t ∈ (0, π 2 ), the second inequality in (6) follows from the right hand side of the Jordan inequality in [2, Section 2.3] and the papers [1,3,4,7]. When t > π 2 , the second inequality in (6) follows from sin t ≤ 1 on (0, ∞) and standard argument. The double inequality (3) is thus proved. The proof of Theorem 2.1 is complete.

A double inequality for the norm of cosine along a circle
In this section, we find a double inequality for the cosine function.

9)
The left equality is valid if and only if θ = 0, π while the right equality is valid if and only if θ = π 2 , 3π 2 . Proof. It is easy to see that, for fixed r > 0, cos re iθ = | cos r| for θ = 0, π, cos re iθ = cosh r for θ = π 2 , 3π 2 , and cos re iθ has a least positive periodicity π with respect to the argument θ.
In Figure 3, we plot the 3D graph of cos re iθ for r ∈ [0, 5] and θ ∈ [0, 2π). In Figure 4, we plot the graph of cos re iθ for r = π and θ ∈ [0, 2π). These two figures are helpful for analyzing and understanding the behaviour of the cosine function cos z along the circle C(0, r) centered at the origin z = 0 of radius r.
Considering odevity of sinh t and sin t, two inequalities in (10) and (11) are equivalent to sinh t t > 1 and sin t t > −1 for t ∈ (0, ∞). The first inequality in (12) follows from cosh x > 1 for x = 0 and the Lazarević inequality (7). When t ∈ (0, π 2 ), the second inequality in (12) follows from the left hand side of the Jordan inequality (8). When t > π 2 , the second inequality in (12) follows from sin t ≥ −1 on (0, ∞) and simple argument. The double inequality (9) is thus proved. The proof of Theorem 3.1 is complete.

Remarks
From Figures 1 and 3, it is not easy to see the difference between sin re iθ and cos re iθ . In fact, the difference sin re iθ − cos re iθ for r ∈ [0, 2π] and θ ∈ [0, 2π) can be showed by Figure 5. From Figures 2 and 4, it is not easy to see the difference between sin πe iθ and cos πe iθ . In fact, the difference sin πe iθ − cos πe iθ for θ ∈ [0, 2π) can be demonstrated by Figure 6.