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Abstract
In the present paper, we obtain certain sufficient conditions for mero-
morphic p-valent functions. Several corollaries and consequences of the main results
are also considered.
Key Words: Meromorphic multivalent functions, meromorphic starlike func-
tions, meromorphic convex functions, meromorphic close-to-convex functions.
Contents
1
Results & Lemmas (12)
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. Let the (non constant) function w(z) be analytic in U with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r <…
Lemma 1.1. Let the (non constant) function w(z) be analytic in U with w(0) = 0. If |w(z)| attains its maximum value on the circle |z| = r < 1 at a point z0 ∈U, then z0w′(z0) = mw(z0) where m is a real number and m ≥n where n ≥1. 2. Main Results With the aid of Lemma 1.1, we derive the next two theorems.
Theorem 2.1.
Theorem 2.1. Let the function f ∈Σp, satisfies the inequality −R αzf ′(z) f(z) + β 1 + zf ′′(z) f ′(z) > [2(α + β)p + n] + λ [2(α +…
Theorem 2.1. Let the function f ∈Σp, satisfies the inequality −R αzf ′(z) f(z) + β 1 + zf ′′(z) f ′(z) > [2(α + β)p + n] + λ [2(α + β)p −n] 2(1 + λ) . (2.1) Then R
Theorem 2.2.
Theorem 2.2. Let the function f ∈Σp, satisfies the inequality −R αzf ′(z) f(z) + β 1 + zf ′′(z) f ′(z) < (α + β)p + n λ + 2p(α + β) +…
Theorem 2.2. Let the function f ∈Σp, satisfies the inequality −R αzf ′(z) f(z) + β 1 + zf ′′(z) f ′(z) < {(α + β)p + n} λ + {2p(α + β) + n} λ + 2 . (3.1) Then R "
Corollary 3.1.
Corollary 3.1. If the function f ∈Σp satisfies the inequality −R 1 + zf ′′(z) f ′(z) > (2p + n) + λ (2p −n) 2 (1 + λ) (λ ≥1, p, n ∈N)…
Corollary 3.1. If the function f ∈Σp satisfies the inequality −R 1 + zf ′′(z) f ′(z) > (2p + n) + λ (2p −n) 2 (1 + λ) (λ ≥1, p, n ∈N) then R −zp+1f ′(z) p > 1 + λ
Corollary 3.2.
Corollary 3.2. If the function f ∈Σ satisfies the inequality −R 1 + zf ′′(z) f ′(z) > 3 + λ 2 (1 + λ) (λ ≥1) then R −z2f ′(z)
Corollary 3.2. If the function f ∈Σ satisfies the inequality −R 1 + zf ′′(z) f ′(z) > 3 + λ 2 (1 + λ) (λ ≥1) then R −z2f ′(z)
Corollary 3.3.
Corollary 3.3. Let the function f ∈Σp, satisfies the inequality −R zf ′(z) f(z) > (2p + n) + λ (2p −n) 2 (1 + λ) (λ ≥1, p, n ∈N). Then R…
Corollary 3.3. Let the function f ∈Σp, satisfies the inequality −R zf ′(z) f(z) > (2p + n) + λ (2p −n) 2 (1 + λ) (λ ≥1, p, n ∈N). Then R (zpf(z)) > 1 + λ 2 . Setting p = n = 1 in Corollary 3.3, the result reduces to
Corollary 3.4.
Corollary 3.4. Let the function f ∈Σ, satisfies the inequality −R zf ′(z) f(z) > 3 + λ 2(1 + λ) (λ ≥1). Then R (zf(z)) > 1 + λ 2.
Corollary 3.4. Let the function f ∈Σ, satisfies the inequality −R zf ′(z) f(z) > 3 + λ 2(1 + λ) (λ ≥1). Then R (zf(z)) > 1 + λ 2 .
Corollary 3.5.
Corollary 3.5. Let the function f ∈Σp, satisfies the inequality −R (1 −γ)zf ′(z) f(z) + γ 1 + zf ′′(z) f ′(z) > p + n 2 1 −λ 1 + λ …
Corollary 3.5. Let the function f ∈Σp, satisfies the inequality −R (1 −γ)zf ′(z) f(z) + γ 1 + zf ′′(z) f ′(z) > p + n 2 1 −λ 1 + λ (λ ≥1, p, n ∈N).
Corollary 3.6.
Corollary 3.6. If the function f ∈Σp satisfies the inequality −R 1 + zf ′′(z) f ′(z) < (p + n)λ + (2p + n) λ + 2 (λ ≥1, p, n ∈N) then R…
Corollary 3.6. If the function f ∈Σp satisfies the inequality −R 1 + zf ′′(z) f ′(z) < (p + n)λ + (2p + n) λ + 2 (λ ≥1, p, n ∈N) then R −zp+1 p f ′(z)
Corollary 3.7.
Corollary 3.7. If the function f ∈Σ satisfies the inequality −R 1 + zf ′′(z) f ′(z) < 2λ + 3 λ + 2 (λ ≥1) then R −z2f ′(z) > 1
Corollary 3.7. If the function f ∈Σ satisfies the inequality −R 1 + zf ′′(z) f ′(z) < 2λ + 3 λ + 2 (λ ≥1) then R −z2f ′(z) > 1
Corollary 3.8.
Corollary 3.8. Let the function f ∈Σp, satisfies the inequality −R zf ′(z) f(z) < (p + n)λ + (2p + n) λ + 2 (λ ≥1, p, n ∈N). Then R…
Corollary 3.8. Let the function f ∈Σp, satisfies the inequality −R zf ′(z) f(z) < (p + n)λ + (2p + n) λ + 2 (λ ≥1, p, n ∈N). Then R [(zpf(z))] > 1 2 + λ.
Corollary 3.9.
Corollary 3.9. Let the function f ∈Σ, satisfies the inequality −R zf ′(z) f(z) < 3 + 2λ 2 + λ (λ ≥1). Then R [(zf(z))] > 1 2 + λ.…
Corollary 3.9. Let the function f ∈Σ, satisfies the inequality −R zf ′(z) f(z) < 3 + 2λ 2 + λ (λ ≥1). Then R [(zf(z))] > 1 2 + λ. Acknowledgments The authors would like to thank the referee for his helpful comments and sug- gestions.
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