By Árpád Baricz (auth.)

In this quantity we examine the generalized Bessel capabilities of the 1st variety by utilizing a few classical and new findings in advanced and classical research. Our target is to provide fascinating geometric houses and practical inequalities for those generalized Bessel capabilities. in addition, we expand many recognized inequalities related to round and hyperbolic services to Bessel and converted Bessel functions.

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**Extra info for Generalized Bessel Functions of the First Kind**

**Sample text**

In addition to the above condition, in his original paper Z. Nehari [158] showed that S f (z) ≤ π 2 /2 for all z ∈ D is also a sufficient condition for the function f to be univalent in D. Here the constant π 2 /2 is sharp. However, Nehari’s conditions may be awkward to verify because it requires the computation of the Schwarzian derivative. Thus, it is often simpler to work directly with the logarithmic derivative f / f of f , called sometimes as the pre-Schwarzian derivative. Such univalence criterion, which involves the pre-Schwarzian derivative is due to J.

For convenience we just sketch the proof. 3 implies Re u p+1 (z) > 0 for all z ∈ D. 2 it follows that u p (z) = 0 for all z ∈ D. Define h : D → C by h(z) = 1 + zu p (z) . u p (z) The function h is analytic in D and h(0) = 1. 11 it is shown that h satisfies the differential equation 4zh (z) + 4h2 (z) + 4(κ − 2)h(z) + cz − 4(κ − 1) = 0. 19) implies ψ (h(z), zh (z); z) ∈ E for all z ∈ D. 5 to prove that Re h(z) > 0 for all z ∈ D. Let z = x + iy ∈ D and c = c1 + ic2 (with x, y, c1 , c2 ∈ R). For all ρ , σ ∈ R satisfying σ ≤ −(1 + ρ 2 )/2 we obtain Re ψ (ρ i, σ ; x + iy) = 4σ − 4ρ 2 − 4ρ Im κ + c1 x − c2 y − 4(Re κ − 1) ≤ −6ρ 2 − 4(Im κ )ρ + c1 x − c2y − 2(2 Re κ − 1) = Q1 (ρ ).

Condition zu p (z) 1 − 3α /2 + α 2 , < u p (z) 1−α where α ∈ [0, 1/2] and z ∈ D, then it is starlike of order α with respect to 1. Proof. We define the function h : D → C by h(z) = [u p (z) − b0 ]/b1 . Then h ∈ A and zu p (z) 1 − 3α /2 + α 2 zh (z) , = < h (z) u p (z) 1−α where α ∈ [0, 1/2] and z ∈ D. e. h is starlike of order α with respect to the origin for α ∈ [0, 1/2]. 9 follows from the definition of the function h, because b0 = 1. 5. 9. 8, we have the following result on the convexity of the generalized Bessel functions.