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Abstract
In this paper, it is aimed to determine the radii of starlikeness and convexity of the normalized generalized Struve functions for three different kinds of normalization and to find tight lower and upper bounds for the radius of starlikeness and convexity of these normalized Struve functions by making use of Euler-Rayleigh inequalities. The Laguerre-Pólya class of entire functions has a crucial role in constructing our main results.
Results & Lemmas (17)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.1
Lemma 1.1 ( see [15]). If a > b > r ≥|z|, and λ ∈[0, 1], then (1.2)
Lemma 1.1 ( see [15]). If a > b > r ≥|z| , and λ ∈[0, 1], then (1.2)
Lemma 1.2
Lemma 1.2 (Runckel, 1969). If f(z) = P n≥0 anzn can be represented by f(z) = eaz2h(z), where a ≤0 and h is of form h(z) = cebz Y n≥1 1 −z…
Lemma 1.2 (Runckel, 1969). If f(z) = P n≥0 anzn can be represented by f(z) = eaz2h(z), where a ≤0 and h is of form h(z) = cebz Y n≥1 1 −z cn e z cn , a, b ∈R, X n≥1
Lemma 2.1.
Lemma 2.1. Let δ, q, b, c > 0 and p + 1 > 0. Then z 7→qWp,b,c,δ(z) possesses infinitely many zeros which are all real. Denoting by qωp,b,c,n…
Lemma 2.1. Let δ, q, b, c > 0 and p + 1 > 0. Then z 7→qWp,b,c,δ(z) possesses infinitely many zeros which are all real. Denoting by qωp,b,c,n the nth positive zero of qWp,b,c,δ(z), under the same conditions the Weierstrassian decomposition (2.1) 2p+1Γ(p δ + b + 2 2 )qWp,b,c,δ(z) = zp+1 Y n≥1
Theorem 2.1. · radius
Theorem 2.1. Let δ, q, b, c > 0, p + 1 > 0 and α ∈[0, 1). Then the following assertions are true. a. The radius of starlikeness of order α…
Theorem 2.1. Let δ, q, b, c > 0, p + 1 > 0 and α ∈[0, 1) . Then the following assertions are true. a. The radius of starlikeness of order α of the function qfp,b,c,δ is r⋆ α(qfp,b,c,δ) = qxp,b,c,δ,1, where qxp,b,c,δ,1 is the smallest zero of the equation rqW ′ p,b,c,δ(r) −α(p + 1)qWp,b,c,δ(r) = 0.
Theorem 2.2. · radius
Theorem 2.2. Let δ, q, b, c > 0 and p + 1 > 0. The radius of starlikeness r⋆(qfp,b,c,δ) is satisfies v u u t 2(p + 1)Γ(q + p δ + b+2 2 ) c(p…
Theorem 2.2. Let δ, q, b, c > 0 and p + 1 > 0. The radius of starlikeness r⋆(qfp,b,c,δ) is satisfies v u u t 2(p + 1)Γ(q + p δ + b+2 2 ) c(p + 3)Γ( p δ + b+2 2 ) < r⋆(qfp,b,c,δ) < 2 v u u u
Theorem 2.3. · radius
Theorem 2.3. Let δ, q, b, c > 0 and p + 1 > 0. The radius of starlikeness r⋆(qgp,b,c,δ) is satisfies 2 s Γ(q + p δ + b+2 2 ) 3cΓ( p δ + b+2…
Theorem 2.3. Let δ, q, b, c > 0 and p + 1 > 0. The radius of starlikeness r⋆(qgp,b,c,δ) is satisfies 2 s Γ(q + p δ + b+2 2 ) 3cΓ( p δ + b+2 2 ) < r⋆(qgp,b,c,δ) < 2 s 3Γ(q + p δ + b+2 2 )Γ(2q + p δ + b+2 2 )
Theorem 2.4. · radius
Theorem 2.4. The radius of starlikeness r⋆(qhp,b,c,δ) is satisfies 2Γ(q + p δ + b+2 2 ) cΓ( p δ + b+2 2 ) < r⋆(qhp,b,c,δ) < 8Γ(q + p δ + b+2…
Theorem 2.4. The radius of starlikeness r⋆(qhp,b,c,δ) is satisfies 2Γ(q + p δ + b+2 2 ) cΓ( p δ + b+2 2 ) < r⋆(qhp,b,c,δ) < 8Γ(q + p δ + b+2 2 )Γ(2q + p δ + b+2 2 ) c
Theorem 2.5. · radius
Theorem 2.5. Let δ, q, b, c > 0, p + 1 > 0 and α ∈[0, 1). Then the following assertions hold true. a. The radius of convexity of order α of…
Theorem 2.5. Let δ, q, b, c > 0, p + 1 > 0 and α ∈[0, 1) . Then the following assertions hold true. a. The radius of convexity of order α of the function qfp,b,c,δ is the smallest root of the equation 1 + 1 p + 1 −1 r qW ′ p,b,c,δ(r) qWp,b,c,δ(r) + r qW ′′ p,b,c,δ(r) qW ′ p,b,c,δ(r) = α.
Theorem 2.6. · radius
Theorem 2.6. Let δ, q, b, c > 0 and p + 1 > 0. Then the radius of convexity rc(qgp,b,c,δ) of the function z 7→qgp,b,c,δ(z) = 2p+1Γ(p δ + b…
Theorem 2.6. Let δ, q, b, c > 0 and p + 1 > 0. Then the radius of convexity rc(qgp,b,c,δ) of the function z 7→qgp,b,c,δ(z) = 2p+1Γ(p δ + b + 2 2 )z−p qWp,b,c,δ(z), is the smallest root of the zqg′ p,b,c,δ ′ = 0 and satisfies the following inequality 2 3 s
Theorem 2.3.
Repeated OCR label — verify against source
Theorem 2.3. Suppose that q̺p,b,c,δ,n’s are the positive zeros of the function qΛp,b,c,δ. Then the function qΛp,b,c,δ has the infinite…
Theorem 2.3. Suppose that q̺p,b,c,δ,n’s are the positive zeros of the function qΛp,b,c,δ. Then the function qΛp,b,c,δ has the infinite product representation as follows: (2.11) qΛp,b,c,δ(z) = Y n≥1 1 − z q̺p,b,c,δ,n . Having taken the logarithmic derivation of the above equality, we obtain (2.12) qΛ′
Theorem 2.7. · radius
Theorem 2.7. Let δ, q, b, c > 0 and p + 1 > 0. Then the radius of convexity rc(qhp,b,c,δ) of the function z 7→qhp,b,c,δ(z) = 2p+1Γ(p δ + b…
Theorem 2.7. Let δ, q, b, c > 0 and p + 1 > 0. Then the radius of convexity rc(qhp,b,c,δ) of the function z 7→qhp,b,c,δ(z) = 2p+1Γ(p δ + b + 2 2 )z1−p+1 2 qWp,b,c,δ(√z)
Theorem 2.6 · radius
Repeated OCR label — verify against source
Theorem 2.6) one can deduce that the radius of convexity rc(qhp,b,c,δ) is the smallest positive root of the equation zqh′ p,b,c,δ(z) ′ =…
Theorem 2.6) one can deduce that the radius of convexity rc(qhp,b,c,δ) is the smallest positive root of the equation zqh′ p,b,c,δ(z) ′ = 0. By taking √z instead of z in (2.8) and (2.9), respectively we can draw a conclusion (2.14) qλp,b,c,δ(z) = zqh′ p,b,c,δ(z) ′ = 1 + Γ( p δ + b+2 2 )
Corollary 2.1. · radius
Corollary 2.1. Let ν > −1 and α ∈[0, 1). a. The radius of starlikeness of order α of 1fν−1,2,1,1(z) = [2νΓ(ν + 1)Jν(z)] 1 ν is the smallest…
Corollary 2.1. Let ν > −1 and α ∈[0, 1) . a. The radius of starlikeness of order α of 1fν−1,2,1,1(z) = [2νΓ(ν + 1)Jν(z)] 1 ν is the smallest positive root of the equation zJ′ ν(z) −ανJν(z) = 0.
Corollary 2.2. · radius
Corollary 2.2. If ν > −1, then the radius of starlikeness r⋆(1fν−1,2,1,1) satisfies 2 r ν(ν + 1) ν + 2 < r⋆(1fν−1,2,1,1) < 2(ν + 2) r ν(ν +…
Corollary 2.2. If ν > −1, then the radius of starlikeness r⋆(1fν−1,2,1,1) satisfies 2 r ν(ν + 1) ν + 2 < r⋆(1fν−1,2,1,1) < 2(ν + 2) r ν(ν + 1) ν2 + 8ν + 8. It is obvious that our main results which are presented in Theorem 2.3 and Theorem 2.4 when we take q = 2, p = ν −1, b = 2, c = 1 and δ = 1, coincide with the inequalities in [3, Thm. 1] and [3, Thm. 2], respectively.
Corollary 2.3. · radius
Corollary 2.3. Let ν > −1. The following assertions hold true: a. The radius of starlikeness r⋆(1gν−1,2,1,1) satisfies 2 r ν + 1 3 <…
Corollary 2.3. Let ν > −1. The following assertions hold true: a. The radius of starlikeness r⋆(1gν−1,2,1,1) satisfies 2 r ν + 1 3 < r⋆(1gν−1,2,1,1) < 2 r 3(ν + 1)(ν + 2) 4ν + 13 . b. The radius of starlikeness r⋆(1hν−1,2,1,1) satisfies 2(ν + 1) < r⋆(1hν−1,2,1,1) < 8(ν + 1)(ν + 2) ν + 5 .
Corollary 2.4. · radius
Corollary 2.4. Let ν > −1 and α ∈[0, 1). Then the following assertions hold true a. The radius of convexity of order α of the function…
Corollary 2.4. Let ν > −1 and α ∈[0, 1) . Then the following assertions hold true a. The radius of convexity of order α of the function 1fν−1,2,1,1 is the smallest positive root of the equation 1 + rJ′′ ν (r) J′ν(r) + 1 ν −1 rJ′ ν(r) Jν(r) = α. b. The radius of convexity of order α of the function 1gν−1,2,1,1 is the smallest positive root of the equation 1 + rJν+2(r) −3Jν+1(r) Jν(r) −rJν+1(r)
Corollary 2.5. · radius
Corollary 2.5. Let ν > −1. The following assertions hold true: a. The radius of convexity rc(1gν−1,2,1,1) satisfies the following…
Corollary 2.5. Let ν > −1. The following assertions hold true: a. The radius of convexity rc(1gν−1,2,1,1) satisfies the following inequality: 2√ν + 1 3 < rc(1gν−1,2,1,1) < 6 r (ν + 1)(ν + 2) 56ν + 137 . b. The radius of convexity rc(1hν−1,2,1,1) satisfies the following inequality: ν + 1 < rc(1hν−1,2,1,1) < 16(ν + 1)(ν + 2) 7ν + 23 .
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