Results & Lemmas (8)
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Theorem 1. · radius
Theorem 1. Let αi > −1 for i ∈ 1, 2,..., d. Then the radius of starlikeness r⋆(fαd) of the normalized hyper-Bessel function z 7→fαd(z) =…
Theorem 1. Let αi > −1 for i ∈{1, 2, . . ., d}. Then the radius of starlikeness r⋆(fαd) of the normalized hyper-Bessel function z 7→fαd(z) = zJαd(z) is the smallest positive root of the equation zJ ′ αd(z) + Jαd(z) = 0 and satisfies the following inequalities (2.1) r⋆(fαd) < d+1q (d + 1)d A(αd) and (2.2) (d + 1)d+1 A(αd) d + 2 < (r⋆(fαd))d+1 < (d + 2) (d + 1)d+1 A(αd)B(αd)
Theorem 2. · radius
Theorem 2. Let αi > −1 for i ∈ 1, 2,..., d. Then the radius of convexity rc (fαd) of the normalized hyper-Bessel function z 7→fαd(z) =…
Theorem 2. Let αi > −1 for i ∈{1, 2, . . ., d}. Then the radius of convexity rc (fαd) of the normalized hyper-Bessel function z 7→fαd(z) = zJαd(z) is the smallest positive root of the equation Jαd(z) + 3zJ ′ αd(z) + z2J ′′ αd(z) = 0 and satisfies the following inequality (2.3) (d + 1)d+1 A(αd) (d + 2)2 < (rc (fαd))d+1 < (d + 1)d+1 (d + 2)2 A(αd)B(αd) (d + 2)4 B(αd) −(2d + 3)2 A(αd) . Here, it is important to note that if we take d = 1 and α1 = ν in the inequality (2.3) we get the inequality [2, T
Theorem 3. · radius
Theorem 3. Let αi > −1 for i ∈ 1, 2,..., d. Then the radius of uniform convexity ruc (fαd) of the normalized hyper-Bessel function z…
Theorem 3. Let αi > −1 for i ∈{1, 2, . . ., d}. Then the radius of uniform convexity ruc (fαd) of the normalized hyper-Bessel function z 7→fαd(z) = zJαd(z) is the smallest positive root of the equation 2z2J ′′ αd(z) + 5zJ ′ αd(z) + Jαd(z) = 0. It is worth also to mention that if we take d = 1 and α1 = ν, we reobtain a recent result on the radius of uniform convexity of Bessel functions [20, Theorem 3.2] which state that, if ν > −1, then the radius of uniform convexity of the function z 7→2νΓ (ν
Theorem 4.
Theorem 4. Let αi > −1 for i ∈ 1, 2,..., d. Then, the first positive zero jαd,1 of the hyper-Bessel function z 7→Jαd(z) satisfies following…
Theorem 4. Let αi > −1 for i ∈{1, 2, . . ., d}. Then, the first positive zero jαd,1 of the hyper-Bessel function z 7→Jαd(z) satisfies following inequalities (d + 1)d+1 A(αd) < jd+1 αd,1 < (d + 1)d+1 A(αd)B(αd) B(αd) −A(αd) and (d + 1)d+1 A(αd) s B(αd) B(αd) −A(αd) < jd+1 αd,1 < 2 (d + 1)d+1 A(αd)B(αd)C(αd) −(A(αd))2 C(αd) (A(αd))2 −3A(αd)C(αd) + 2B(αd)C(αd)
Theorem 5.
Theorem 5. Let αi > −1 for i ∈ 1, 2,..., d. Then the zeros of the function z 7→J ′ αd(z) are interlaced with those of the function z…
Theorem 5. Let αi > −1 for i ∈{1, 2, . . ., d}. Then the zeros of the function z 7→J ′ αd(z) are interlaced with those of the function z 7→Jαd(z). 2.6. Redheffer-type inequalities and bounds for normalized hyper-Bessel functions. In this subsection we find Redheffer-type inequalities, monotonicity, and bounds for normalized hyper-Bessel functions.
Theorem 6.
Theorem 6. Let αi > −1 for i ∈ 1, 2,..., d. Then the function hαd, qαd: [0, jd+1 αd,1) →[0, ∞) defined by hαd(x) =
Theorem 6. Let αi > −1 for i ∈{1, 2, . . ., d}. Then the function hαd, qαd : [0, jd+1 αd,1) →[0, ∞) defined by hαd(x) =
Corollary 1.
Corollary 1. Let αi > −1 for i ∈ 1, 2,..., d and x ∈[0, jd+1 αd,1). Then Jαd(x) ≤ x d + 1 S(αd) e − xd+1 (d+1)d+1A(αd) A(αd). Finally,…
Corollary 1. Let αi > −1 for i ∈{1, 2, . . ., d} and x ∈[0, jd+1 αd,1). Then Jαd(x) ≤ x d + 1 S(αd) e − xd+1 (d+1)d+1A(αd) A(αd) . Finally, we obtain some bounds for real hyper-Bessel functions.
Theorem 7.
Theorem 7. If αi > −1 for i ∈ 1, 2,..., d, and x ∈(0, jαd,1) then the following sharp exponential Redheffer-type inequalities hold: (2.4)
Theorem 7. If αi > −1 for i ∈{1, 2, . . ., d}, and x ∈(0, jαd,1) then the following sharp exponential Redheffer-type inequalities hold: (2.4)
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