Geometric Properties of Generalized Bessel Function Associated with the Exponential Function

Adiba Naz; Sumit Nagpal; V. Ravichandran

doi:10.1515/ms-2023-0106

Artikel

Geometric Properties of Generalized Bessel Function Associated with the Exponential Function

Adiba Naz , Sumit Nagpal und V. Ravichandran

Veröffentlicht/Copyright: 18. Dezember 2023

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ABSTRACT

Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind, which is an elementary transform of the hypergeometric function and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz et al. [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. 38(3) (2015), 1255–1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results.

2020 Mathematics Subject Classification: Primary 33C10; 33B10; Secondary 30C45; 30C80

Keywords: Hypergeometric function; special function; Bessel function; modified Bessel function; starlike functions; differential subordination; exponential function

Dedicated to Prof. Ajay Kumar on the occasion of his retirement

(Communicated by Tomasz Natkaniec)

Appendix

Consider the function

h(m,θ):=(mecosθcos(θ+sinθ)+e2cosθcos(2sinθ)−1)2 +(mecosθsin(θ+sinθ)+e2cosθsin(2sinθ))2=m2e2cosθ+2mecosθ(e2costcos(θ−sinθ)−cos(θ+sinθ)) +e4cosθ−2e2cosθcos(2sinθ)+1,

where m ≥ 1 and θ ∈ [0, 2π). Then

∂h∂m=2me2cosθ+2ecosθ(e2cosθcos(θ−sinθ)−cos(θ+sinθ))>0

for all m ≥ 1 and θ ∈ [0, 2π) (see Figure 5). This implies that h is an increasing function of m and hence h(m, θ) ≥ h(1, θ) for all m ≥ 1 and θ ∈ [0, 2π). Now consider the following function

h(1,θ)=e4cosθ+e2cosθ−2e2cosθcos(2sinθ)+2ecosθ(e2costcos(θ−sinθ) −cos(θ+sinθ))+1=:g(θ),

Figure 5.

Graph of the function 2e^{cos θ} (e^{2 cos θ} cos(θ – sin θ) – cos(θ + sin θ))

where θ ∈ [0, 2π). Then

g′(θ)=2ecosθ( sin(t+sint)+sin(2t+sint)+ecost( sin(t+2sint) −ecost(2ecostsint+sin(t−sint)+sin(2t−sint)+2sin(sint)) ) ).

From the graph of the derivative of the transcendental function g with respect to θ and the fact that g′(0) = g′(π) = 0, we observe that 0 and π are the only critical points of g in the interval [0, 2π) (see Figure 6(a)).

Figure 6.

Further, the calculations

g″(0)=2e(5+3e−3e2−2e3)<0andg″(π)=2e4(2+e+e2+e3)>0

imply that g attains a local minimum at θ = π using the second derivative test for one variable (which states that for a twice differentiable function f, if f″(x) < 0, then f has local maximum at x and if f″(x) > 0, then f has local minimum at x, where x is the critical point of f). That minimum value is given by g(π) = (1 – 1/e² + 1/e)². Moreover considering the end points of the interval [0, 2π) proves that the function g attains its global minimum at θ = π (which is also evident from Figure 6(b)).

Similarly, for m ≥ 1, α > 1/e and θ ∈ [0, 2π), let us calculate the minimum value of the function

ℓ(m,θ)=(ecosθcos(sinθ)+αmcosθ−1)2+(ecosθsin(sinθ)+αmsinθ)2=1−2ecosθcos(sinθ)+e2cosθ+m2α2+2mα(ecosθcos(θ−sinθ)−cosθ).

A direct calculation gives

∂ℓ∂m=2mα2+2α(ecosθcos(θ−sinθ)−cosθ)>0

for α > 1/e and θ ∈ [0, 2π) (see Figure 7). This shows that ℓ is also an increasing function of the parameter m, and therefore

ℓ(m,θ)≥ℓ(1,θ)=1−2ecosθcos(sinθ)+e2cosθ+α2 +2α(ecosθcos(θ−sinθ)−cosθ)=:l1(θ).

Figure 7.

Graph of the function e^{cos θ} cos(θ – sin θ) – cos θ

Now,

l1′(θ)=2ecosθcosθsin(sinθ)−2e2cosθsinθ+2ecosθcos(sinθ)sinθ2α(sinθ−ecosθcos(θ−sinθ)sinθ−ecosθ(1−cosθ)sin(θ−sinθ)).

Clearly, l1′(0)=l1′(π)=0 . Also, if we write

l1′(θ)=u(θ)+2αv(θ),

where u(θ) = 2e^cosθ(cos θ sin(sin θ) – e^{cos θ} sin θ + cos(sin θ) sin θ) and v(θ) = sin θ – e^{cos θ} cos(θ – sin θ) sin θ – e^{cos θ}(1 – cos θ) sin(θ – sin θ), then Figure 8 depicts that 0 > u(θ) > v(θ) for θ ∈ (0, π) and 0 < u(θ) < v(θ) for θ ∈ (π, 2π). Consequently, if α > 1/e, it follows that l1′(θ)<(1+2α)u(θ)<0 for θ ∈ (0, π) and l1′(θ)>(1+2α)v(θ)>0 for θ ∈ (π, 2π). Thus the only critical points of the function l₁ in the interval [0, 2π) are 0 and π.

Figure 8.

Graph of the functions u(θ) (dotted) and v(θ)

Again the inequalities

l1′′(0)=−2(e(e−2)+α(e−1))<0

and

l1′′(π)=1e2(2+2α(3−e)e)>0

for α > 1/e imply that the function l₁ attains its local minimum value at θ = π. Furthermore examining the value of l₁ at the end points of the interval [0, 2π) indeed proves that the function l₁ attains global minimum at θ = π and that minimum value is l₁(π) = (α – 1/e + 1)².

Acknowledgement

The authors are thankful to referees for their insightful suggestions.

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Received: 2022-09-04

Accepted: 2023-01-10

Published Online: 2023-12-18

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Artikel in diesem Heft

https://doi.org/10.1515/ms-2023-0106

Schlagwörter für diesen Artikel

Hypergeometric function; special function; Bessel function; modified Bessel function; starlike functions; differential subordination; exponential function