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Geometric Properties of Generalized Bessel Function Associated with the Exponential Function

  • Adiba Naz , Sumit Nagpal EMAIL logo and V. Ravichandran
Published/Copyright: December 18, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 6

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 2ecos θ (e2 cos θ cos(θ – sin θ) – cos(θ + sin θ))
Figure 5.

Graph of the function 2ecos θ (e2 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.
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 ecos θ cos(θ – sin θ) – cos θ
Figure 7.

Graph of the function ecos 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) = 2ecosθ(cos θ sin(sin θ) – ecos θ sin θ + cos(sin θ) sin θ) and v(θ) = sin θ – ecos θ cos(θ – sin θ) sin θ – ecos 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)
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.

REFERENCES

[1] Baricz, A.: Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73(1–2) (2008), 155–178.10.5486/PMD.2008.4126Search in Google Scholar

[2] Baricz, A.: Generalized Bessel Functions of the First Kind, Lecture Notes in Math. 1994, Springer-Verlag, Berlin, 2010.10.1007/978-3-642-12230-9Search in Google Scholar

[3] Baricz, A.—Deniz, E.—Çağlar, M.—Orhan, H.: Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. 38(3) (2015), 1255–1280.10.1007/s40840-014-0079-8Search in Google Scholar

[4] Baricz, A.—Ponnusamy, S.: Starlikeness and convexity of generalized Bessel functions, Integral Transforms Spec. Funct. 21(9–10) (2010), 641–653.10.1080/10652460903516736Search in Google Scholar

[5] Baricz, A.—Szász, R.: The radius of convexity of normalized Bessel functions of the first kind, Anal. Appl. (Singap.) 12(5) (2014), 485–509.10.1142/S0219530514500316Search in Google Scholar

[6] Bohra, N.—Ravichandran, V.: On confluent hypergeometric functions and generalized Bessel functions, Anal. Math. 43(4) (2017), 533–545.10.1007/s10476-017-0203-8Search in Google Scholar

[7] Brown, R. K.: Univalence of Bessel functions, Proc. Amer. Math. Soc. 11 (1960), 278–283.10.1090/S0002-9939-1960-0111846-6Search in Google Scholar

[8] Conway, J. B.: Functions of One Complex Variable, 2nd ed., Graduate Texts in Math. 11, Springer-Verlag, New York, 1978.10.1007/978-1-4612-6313-5_2Search in Google Scholar

[9] Dziok, J.—Srivastava, H. M.: Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103(1) (1999), 1–13.10.1016/S0096-3003(98)10042-5Search in Google Scholar

[10] Kanas, S.—Mondal, S. R.—Mohammed A. D.: Relations between the generalized Bessel functions and the Janowski class, Math. Inequal. Appl. 21(1) (2018), 165–178.10.7153/mia-2018-21-14Search in Google Scholar

[11] Korenev, B. G.: Bessel Functions and their Applications, translated from the Russian by E. V. Pankratiev, Analytical Methods and Special Functions 8, Taylor & Francis Group, London, 2002.10.1201/b12551Search in Google Scholar

[12] Madaan, V.—Kumar, A.—Ravichandran, V.: Lemniscate convexity of generalized Bessel functions, Studia Sci. Math. Hungar. 56(4) (2019), 404–419.10.1556/012.2019.56.4.1445Search in Google Scholar

[13] Mendiratta, R.—Nagpal, S.—Ravichandran, V.: On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38(1) (2015), 365–386.10.1007/s40840-014-0026-8Search in Google Scholar

[14] Miller, S. S.—Mocanu, P. T.: Differential Subordinations: Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics 225, Marcel Dekker, Inc., New York, 2000.10.1201/9781482289817Search in Google Scholar

[15] Mondal, S. R.—Al Dhuain M.: Inclusion of the generalized Bessel functions in the Janowski class, Int. J. Anal. 2016 (2016), Art. ID 4740819.10.1155/2016/4740819Search in Google Scholar

[16] Naz, A.—Nagpal, S.—Ravichandran, V.: Star-likeness associated with the exponential function, Turkish J. Math. 43(3) (2019), 1353–1371.10.3906/mat-1902-34Search in Google Scholar

[17] Petrova, T. S.: Application of Bessels functions in the modelling of chemical engineering processes, Bulgarian Chemical Communications 41 (2009), 343–354.Search in Google Scholar

[18] Prajapat, J. K.: Certain geometric properties of normalized Bessel functions, Appl. Math. Lett. 24(12) (2011), 2133–2139.10.1016/j.aml.2011.06.014Search in Google Scholar

[19] Yan, C. M.—Liu, J. L.: Convolution properties for meromorphically multivalent functions involving generalized Bessel functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 112(1) (2018), 293–299.10.1007/s13398-017-0378-7Search in Google Scholar
Received: 2022-09-04
Accepted: 2023-01-10
Published Online: 2023-12-18

© 2023 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2023-0106
Keywords for this article
Hypergeometric function; special function; Bessel function; modified Bessel function; starlike functions; differential subordination; exponential function

Articles in the same Issue

  1. ℒ-fuzzy Annihilators in Residuated Lattices
  2. Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications
  3. Uniform Local Connectedness and Completion of Metric σ-Frames
  4. On Transcendental Regular Continued Fractions
  5. On Repdigits Which are Sums or Differences of Two k-Pell Numbers
  6. Peirce Decompositions for Evolution Algebras
  7. Generalized Anti-Symmetry Laws in Groupoids
  8. New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications
  9. Geometric Properties of Generalized Bessel Function Associated with the Exponential Function
  10. On the Attainable Set of Iterative Differential Inclusions
  11. On the Differential-Difference Sine-Gordon Equation with an Integral Type Source
  12. A Critical p(x)-Laplacian Steklov Type Problem with Weights
  13. Distributivity of a Uni-nullnorm with Continuous and Archimedean Underlying T-norms and T-conorms Over an Arbitrary Uninorm
  14. Some Approximation Properties of Operators Including Degenerate Appell Polynomials
  15. Operators that are Weakly Coercive and a Compact Perturbation
  16. On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric
  17. Paracompactness and Open Relations
  18. On (Strongly) (Θ-)Discrete Homogeneous Spaces
  19. The Alpha Power Muth-G Distributions and Its Applications in Survival and Reliability Analyses
  20. Weakly Einstein Equivalence in a Golden Space Form and Certain CR-submanifolds
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