Citation: David Igoe, Kenneth Gavin. Characterization of the Blessington sand geotechnical test site[J]. AIMS Geosciences, 2019, 5(2): 145-162. doi: 10.3934/geosci.2019.2.145
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Let Σ(a,k) be the class of meromorphic functions f of the form
f(z)=az+∞∑n=kanzn(a>0,k≥2,k∈N), | (1.1) |
which is analytic in the punctured open unit disk U∗={z∈C:0<|z|<1}=U∖{0}.
In 2009, El-Ashwah [1] introduced the subclass ΣS∗k(a,β) of generalized meromorphically starlike of order β as follows,
ΣS∗k(a,β)={f(z)∈Σ(a,k):−Rezf′(z)f(z)>β,β∈[0,1)}. |
An analytic function g:U={z:|z|<1}→C is subordinate to an analytic function h:U→C, if there is a function ω satisfying ω(0)=0and|ω(z)|<1(z∈U), such that g(z)=h(ω(z))(z∈U). Note that g(z)≺h(z). Especially, if h is univalent in U, then the following conclusion is true (see [2]):
g(z)≺h(z)⟺g(0)=h(0)andg(U)⊂h(U). |
Using the subordinate relationship, we introduce the following subclass of generalized meromorphic Janowski functions (see [3,5,6]).
Definition 1. Let f(z)∈Σ(a,k),a>0,A,B∈R,|A|≤1,|B|≤1 and A≠B. The function f∈ΣS∗k(a,A,B) if and only if
−zf′(z)f(z)≺1+Az1+Bz(z∈U∗). |
It is obvious that
ΣS∗k(a,A,B)⊂ΣS∗k(a,1−A1−B)(−1≤B<A≤1). |
According to the subordination relationship, the function f(z)∈ΣS∗k(a,A,B) if and only if there exists an analytic function ω(z) in the open unit dick U satisfying ω(0)=0 and |ω(z)|<1, such that
−zf′(z)f(z)=1+Aw(z)1+Bw(z)(z∈U∗), |
which is equivalent to
|zf′(z)+f(z)Af(z)+Bzf′(z)|<1(z∈U∗). | (1.2) |
Let T(a,k) be the subclass of Σ(a,k), the function f belonging to T(a,k) of the following form
f(z)=az−∞∑n=k|an|zn. | (1.3) |
Let
¯ΣS∗k(a,A,B)=T(a,k)∩ΣS∗k(a,A,B). |
Next, we introduce the generalized λ-Hadamard product of the class T(a,k).
Definition 2. Let a>0,p>0,q>0,λ≥0,k∈N,k≥2,fi(z)=az−∑∞n=k|an,j|zn∈T(a,k)(i=1,2). The generalized λ-Hadamard product (f1˜△f2)λ(p,q,a;z) of the function f1 and f2 is defined by
(f1˜Δf2)λ(p,q,a;z)=(1−λ)(f1Δf2)(p,q,a;z)−λz(f1Δf2)′(p,q,a;z), | (1.4) |
where
(f1Δf2)(p,q,a;z)=a2z−∞∑n=k|an,1|p|an,2|qzn. | (1.5) |
From (1.4) and (1.5), we get
(f1˜Δf2)λ(p,q,a;z)=a2z−∞∑n=k[1−(n+1)λ]|an,1|p|an,2|qzn. |
Selecting different parameters a,λ,p and q, we can obtain the following three special convolutions:
(ⅰ) For λ=0, (f1˜Δf2)0(p,q,a;z) is the generalized Hadamard product:
(f1˜Δf2)0(p,q,a;z)=:(f1Δf2)(p,q,a;z)=a2z−∞∑n=k|an,1|p|an,2|qzn. |
In particular, for a=1, the Hadamard product (f1Δf2)(p,q,1;z)=1z−∑∞n=k|an,1|p|an,2|qzn was studied by Janowski [3] and Tang et al. [6].
(ⅱ) For λ=0,p=q=1, (f1˜Δf2)0(1,1,a;z) is the general Hadamard product:
(f1˜Δf2)0(1,1,a;z)=:(f1∗f2)(a;z)=a2z−∞∑n=k|an,1||an,2|zn. |
In particular, for a=1, (f1∗f2)(1;z) is the well-known Hadamard product (see [1,4,7]):
(f1∗f2)(1;z)=(f1∗f2)(z)=1z−∞∑n=k|an,1||an,2|zn. |
For a=1,k=1, f1(z)=z−1+∑∞n=1(a)n(c)nzn,f2(z)=az−∑∞n=1|an,2|zn, (a)n=a(a+1)⋯(a+n−1)(n∈N), (f1∗f2)(1;z)=1z−∑∞n=1(a)n(c)n|an,2|zn was studied by Liu et al. [8] (see also [9]).
(ⅲ) For p=q=1, (f1˜Δf2)λ(1,1,a;z)=(f1¯∗f2)λ(z) is λ−Hadamard product:
(f1¯∗f2)λ(z)=(1−λ)(f1∗f2)+λz(f1∗f2)′(z)=a2z−∞∑n=k[1−(n+1)λ]|an,1||an,2|zn. |
In 1996, Choi et al. [4] studied the closure properties of the generalized λ-Hadamard product for univalent functions. In 2001, Liu et al. [8] investigated two new classes of meromorphically multivalent functions by using a linear operator. Based on this, in this paper, we mainly consider the closure properties of the generalized λ-Hadamard product for meromorphic functions. Firstly, we obtain the necessary and sufficient conditions of the class ¯ΣS∗k(a,A,B) by using subordination relationship. Secondly, we discuss the closure properties of the general Hadamard product, the generalized Hadamard product and λ-Hadamard product for functions of the class ¯ΣS∗k(a,A,B). Finally, we discuss the closure properties of the generalized λ-Hadamard product and generalize the known conclusions.
Unless otherwise noted, the parameters a,k,A and B satisfy the condition: a>0,k∈N,k≥2,A,B∈R,|A|≤1,|B|≤1 and A≠B.
In order to establish our results, we need the following lemmas.
Lemma 1. Let f be of the form (1.1). If
∞∑n=k[(n+1)+|A+nB|]|an|≤a|A−B|, | (2.1) |
then f(z)∈ΣS∗k(a,A,B).
Proof. Let |z|=1. By (2.1), we obtain
|zf′(z)+f(z)|−|Af(z)+Bzf′(z)|≤|∞∑n=k(n+1)|an|zn|−|az(A−B)|+|∞∑n=k(A+nB)|an|zn|≤∞∑n=k(n+1)|an|−a|A−B|+∞∑n=k|A+nB||an|=∞∑n=k[(n+1)+|A+nB|]|an|−a|A−B|≤0. |
Using the principle of maximum modulus, we get
|zf′(z)+f(z)|−|Af(z)+Bzf′(z)|<0(z∈U∗). |
Then we conclude that f(z)∈ΣS∗k(a,A,B). Therefore, we complete the proof of Lemma 1.
Lemma 2. Let the function f be of the form (1.3).
(ⅰ) If 0≤B<A≤1. Then the function f(z)∈¯ΣS∗k(a,A,B) if and only if
∞∑n=k[(n+1)+(A+nB)]|an|≤a(A−B). | (2.2) |
(ⅱ) If −1≤A<B≤0. Then the function f(z)∈¯ΣS∗k(a,A,B) if and only if
∞∑n=k[(n+1)−(A+nB)]|an|≤a(B−A). | (2.3) |
Proof. It appears from (1.3) that ¯ΣS∗k(a,A,B)⊂ΣS∗k(a,A,B). In view of Lemma 1, we just need prove the necessity part of Lemma 2. Let f(z)∈¯ΣS∗k(a,A,B). By (1.2), we have
|zf′(z)+f(z)Af(z)+Bzf′(z)|=|∑∞n=k(n+1)|an|zn−1a(A−B)−∑∞n=k(A+nB)|an|zn−1|<1. |
Since |Rez|≤|z| holds true for all z∈C. If 0≤B<A≤1, then
Re{∑∞n=k(1+n)|an|zn−1a(A−B)−∑∞n=k(A+nB)|an|zn−1}<1. | (2.4) |
Let z=r→1−. From (2.4), we can get (2.2). Using the same method, we can obtain the result of (ⅱ). The estimates of (ⅰ) and (ⅱ) are sharp for the function f given by
f(z)=az−a|A−B|(k+1)+|A+kB|zk,k≥2. | (2.5) |
So, we complete the proof of Lemma 2.
First of all, we discuss the closure properties of the general Hadamard product of the class ¯ΣS∗k(a,A,B).
Theorem 1. Let fi(z)=az−∑∞n=k|an|zn∈¯ΣS∗k(a,A,B)(i=1,2). If A and B satisfy the condition
0≤B<A≤1or−1≤A<B≤0, |
then 1a(f1∗f2)(a;z)∈¯ΣS∗k(a,A,B).
Proof. Let 0≤B<A≤1. If fi(z)∈¯ΣS∗k(a,A,B)(i=1,2), from (2.2), we obtain
∞∑n=k[(n+1)+(A+nB)]|an,1|≤a(A−B) | (3.1) |
and
∞∑n=k[(n+1)+(A+nB)]|an,2|≤a(A−B). | (3.2) |
To obtain 1a(f1∗f2)(a;z)∈¯ΣS∗k(a,A,B), we just prove
∞∑n=k1a[(n+1)+(A+nB)]|an,1||an,2|≤a(A−B). | (3.3) |
By using Cauchy-Schwarz inequality, from (3.1) and (3.2), we have
∞∑n=k[(n+1)+(A+nB)]√|an,1||an,2|≤a(A−B). | (3.4) |
According to (3.3) and (3.4), we only need to prove
√|an,1||an,2|≤a. |
From (3.4) and the condition 0≤B<A≤1, we have
√|an,1||an,2|≤a(A−B)(n+1)+(A+nB)≤a. |
Therefore, we conclude that 1a(f1∗f2)(a;z)∈¯ΣS∗k(a,A,B).
For −1≤A<B≤0, using the same method above, we get 1a(f1∗f2)(a;z)∈¯ΣS∗k(a,A,B). Therefore, we complete the proof of Theorem 1.
Next, we prove the closure properties of the generalized Hadamard product.
Theorem 2. Let fi(z)=az−∑∞n=k|an|zn∈¯ΣS∗k(a,A,B)(i=1,2) and p>1. If anyone of the following conditions holds true:
(ⅰ) For 0≤B<A≤1, ˆB and a satisfy
{0≤ˆB≤aA[(k+1)+(A+kB)]−(A−B)(k+1+A)a[(k+1)+(A+kB)]+(A−B)k,0<a<1,ˆB≤aA(1+B)−(A−B)a(1+B)+(A−B),a>1. |
(ⅱ) For −1≤A<B≤0, −1≤A<ˆB≤0 and a satisfies a>0. Then 1a(f1Δf2)(1p,p−1p;a,z)∈¯ΣS∗k(a,A,ˆB).
Proof. (ⅰ) For 0≤B<A≤1, by Lemma 2, we can get
∞∑n=k(n+1)+(A+nB)a(A−B)|an,i|≤1(i=1,2). |
We deduce that
{∞∑n=k(n+1)+(A+nB)a(A−B)|an,1|}1p≤1 | (3.5) |
and
{∞∑n=k(n+1)+(A+nB)a(A−B)|an,2|}p−1p≤1. | (3.6) |
According to (1.5), we have
(f1Δf2)(1p,p−1p)=a2z−∞∑n=k|an,1|1p|an,2|p−1pzn. |
To prove 1a(f1∗f2)(1p,p−1p,a;z)∈¯ΣS∗k(a,A,ˆB), we only need to show
∞∑n=k(n+1)+(A+nˆB)a2(A−ˆB)|an,1|1p|an,2|p−1p≤1. | (3.7) |
Applying Hölder inequality, from (3.5) and (3.6), we obtain
∞∑n=k(n+1)+(A+nB)a(A−B)|an,1|1p|an,2|p−1p≤1. |
Thus, (3.7) holds if
(n+1)+(A+nˆB)a2(A−ˆB)≤(n+1)+(A+nB)a(A−B)(∀n∈N,n≥k), |
that is,
ˆB[a[(n+1)+(A+nB)]A−B+n]≤aA[(n+1)+(A+nB)]A−B−(n+1+A). | (3.8) |
Let
F1(n)=a[(n+1)+(A+nB)]A−B+n |
and
G1(n)=aA[(n+1)+(A+nB)]A−B−(n+1+A). |
It is easy to see F1(n)>0 holds true for n≥k. To obtain (3.8), we only need to prove
ˆB≤G1(n)F1(n)=aA[(n+1)+(A+nB)]−(A−B)(n+1+A)a[(n+1)+(A+nB)]+(A−B)n. |
Because G1(n)F1(n) is an increasing function with respect to n and the condition (i) holds true, so we have
ˆB≤aA[(k+1)+(A+kB)]−(A−B)(k+1+A)a[(k+1)+(A+kB)]+(A−B)k=G(k)F(k)≤G(n)F(n). |
Therefore, we conclude that 1a(f1Δf2)(1p,p−1p;a,z)∈¯ΣS∗k(a,A,ˆB).
(ⅱ) It is similar to the proof of (ⅰ). For −1≤A<B≤0, we only need to show
(n+1)−(A+nˆB)a2(ˆB−A)≤(n+1)−(A+nB)a(B−A)(∀n∈N,n≥k), |
that is,
ˆB[a[(n+1)−(A+nB)]B−A+n]≥aA[(n+1)−(A+nB)]B−A+(n+1−A). | (3.9) |
Let
F2(n)=a[(n+1)−(A+nB)]B−A+n |
and
G2(n)=aA[(n+1)−(A+nB)]B−A+(n+1−A). |
It is easy to see F2(n)>0 holds true for n≥k. To make (3.9) establish, we only need to prove
ˆB≥G2(n)F2(n)=aA[(n+1)−(A+nB)]+(B−A)(n+1−A)a[(n+1)+(A+nB)]+(B−A)n. | (3.10) |
It is clear that G2(n)F2(n) is an decreasing function with respect to n for 0<a<1 and
A>aA[(k+1)−(A+kB)]−(B−A)(k+1−A)a[(k+1)−(A+kB)]+(B−A)k, |
then we have
ˆB≥aA[(k+1)−(A+kB)]−(B−A)(k+1−A)a[(k+1)−(A+kB)]+(B−A)k=G2(k)F2(k)≥G2(n)F2(n). |
It can be concluded that 1a(f1Δf2)(1p,p−1p;a,z)∈¯ΣS∗k(a,A,ˆB). Clearly, G2(n)F2(n) is an increasing function with respect to n for a>1 and
limn→+∞G2(n)F2(n)=aA(1−B)+(B−A)a(1−B)+(B−A). |
Since A>aA(1−B)+(B−A)a(1−B)+(B−A), we have
ˆB>A>aA(1−B)+(B−A)a(1−B)+(B−A)=limn→+∞G2(n)F2(n)≥G2(n)F2(n). |
Thus, we conclude that 1a(f1Δf2)(1p,p−1p;a,z)∈¯ΣS∗k(a,A,ˆB). The proof of Theorem 2 is completed.
Putting A=1,B=0 in Theorem 2, we obtain the following corollary.
Corollary 1. Let fi(z)=az−∑∞n=k|an,i|zn∈¯ΣS∗k(a,1,0)(i=1,2) and p>1. If ˆB and a satisfy
{0≤ˆB≤(a−1)(k+2)a(k+2)+k,0<a<1,ˆB≤a−1a+1,a>1. |
Then 1a(f1△f2)(1p,p−1p;a,z)∈¯ΣS∗k(a,A,ˆB). Finally, we consider the closure properties of λ-Hadamard product (f1¯∗f2)λ(z) and the generalized λ-Hadamard product (f1˜Δf2)λ(p.q,a;z) as follows.
Theorem 3. Let fi(z)=az−∑∞n=k|an,i|zn∈¯ΣS∗k(a,A,B)(i=1,2). If A,B and λ satisfy
{(A−B)−(k+1−A−nB)(A−B)(k+1)<λ<1k+1,0≤B<A≤1,λ<1k+1,−1≤A<B≤0. |
Then 1a(f1¯∗f2)λ(z)∈¯ΣS∗k(a,A,B).
Proof. Let fi(z)∈¯ΣS∗k(a,A,B)(i=1,2). If 0≤B<A≤1, from Lemma 2, we obtain
∞∑n=k((n+1)+(A+nB))|an,1|≤a(A−B), | (3.11) |
and
∞∑n=k((n+1)+(A+nB))|an,2|≤a(A−B). | (3.12) |
By Definition 2, we have
1a(f1ˉ∗f2)λ(z)=az−∞∑n=k1a(1−(n+1)λ|an,1||an,2|zn. |
To show 1a(f1ˉ∗f2)λ(z)∈¯ΣS∗k(a,A,B), we just prove
∞∑n=k1a(1−(n+1)λ)((n+1)+(A+nB))|an,1||an,2|≤a(A−B). | (3.13) |
Applying Cauchy-Schwarz inequality to (3.11) and (3.12), we get
∞∑n=k((n+1)+(A+nB))√|an,1||an,2|≤a(A−B). | (3.14) |
According to (3.13) and (3.14), we only need to prove
√|an,1||an,2|≤a1−(n+1)λ. |
Since √|an,1||an,2|≤a(A−B)(n+1)+(A+nB), we only need to prove
A−B≤(n+1)+(A+nB)1−(n+1)λ,n≥k. | (3.15) |
Let
M(n)=(n+1)+(A+nB)1−(n+1)λ. |
Clearly, M(n) is an increasing function for n≥k. This implies that (3.15) holds true if
k+1+A+nB1−(k+1)λ≥A−B. |
For λ<1k+1, the above formula can be expressed as λ≥(A−B)−(k+1−A−nB)(A−B)(k+1).
To summarize, we conclude that 1a(f1¯∗f2)λ(z)∈¯ΣS∗k(a,A,B) for (A−B)−(k+1−A−nB)(A−B)(k+1)<λ<1k+1. Using the same method, we can get 1a(f1¯∗f2)λ(z)∈¯ΣS∗k(a,A,B) for
λ<min{(B−A)−(k+1−A−nB)(B−A)(k+1),1k+1}=1k+1. |
Thus, we complete the proof of Theorem 3.
Theorem 4. Let fi(z)=az−∑∞n=k|an|zn∈¯ΣS∗k(a,A,B)(i=1,2) and p>1.
(1) For 0≤B<A≤1,0≤ˆB<A. If anyone of the following conditions holds true.
(ⅰ) a<1,λ<min{12(n+1),a(1+B)+(A−B)(2n+1)(A−B),k(A−B)+a(k+1+A+kB)k(k+1)(A−B)},
(ⅱ) a>1,12(k+1)<λ<min{a(1+B)+(A−B)(2n+1)(A−B),k(A−B)+a(k+1+A+kB)k(k+1)(A−B)},
(ⅲ) a<1,max{a(1+B)+(A−B)(2k+1)(A−B),k(A−B)+a(k+1+A+kB)k(k+1)(A−B)}<λ<12(n+1),
(ⅳ) a>1,max{12(k+1),a(1+B)+(A−B)(2k+1)(A−B),k(A−B)+a(k+1+A+kB)k(k+1)(A−B)}<λ.
Then the generalized λ-Hadamard product: 1a(f1˜Δf2)λ(1p,p−1p,a;z)∈¯ΣS∗k(a,A,B).
(2) For −1≤A<B≤0. If anyone of the following conditions holds true.
(ⅴ) a<1,λ<min{12(n+1),(B−A)+a(1−B)(2k−1)(B−A),k(B−A)+a(k+1−A−kB)k(k+1)(B−A)},
max(A,aA[(k+1)−(A+kB)]+[1−(k+1)λ](k+1+A)(B−A)a[(k+1)−(A+kB)]−k[1−(k+1)λ](B−A))≤ˆB≤0,
(ⅵ) a>1,12(k+1)<λ<min{(B−A)+a(1−B)(2k−1)(B−A),k(B−A)+a(k+1−A−kB)k(k+1)(B−A)},
max(A,aA[(k+1)−(A+kB)]+[1−(k+1)λ](k+1+A)(B−A)a[(k+1)−(A+kB)]−k[1−(k+1)λ](B−A))≤ˆB≤0,
(ⅶ) a<1,max{(B−A)+a(1−B)(2k−1)(B−A),k(B−A)+a(k+1−A−kB)k(k+1)(B−A)}<λ<12(n+1),A≤ˆB<0,
(ⅷ) a>1,max{12(k+1),(B−A)+a(1−B)(2k−1)(B−A),k(B−A)+a(k+1−A−kB)k(k+1)(B−A)}<λ,A≤ˆB<0.
Then the generalized λ-Hadamard product: 1a(f1˜Δf2)λ(1p,p−1p,a;z)∈¯ΣS∗k(a,A,B).
Proof. Let fi(z)∈¯ΣS∗k(a,A,B)(i=1,2). The method for the proof of Theorem 4 is similar to that of Theorem 2. If −1≤A<B≤0, we just need to prove
[1−(n+1)λ][(n+1)−(A+nˆB)]a2(A−ˆB)≤(n+1)−(A−nB)a(B−A), | (3.16) |
that is
ˆB[a[(n+1)−(A+nB)]B−A+n(1−(n+1)λ)]≤aA[(n+1)−(A+nB)]B−A+[1−(n+1)λ](n+1−A). |
Let
P(n)=a[(n+1)−(A+nB)]B−A+n[1−(n+1)λ] |
and
Q(n)=aA[(n+1)+(A+nB)]B−A+[1−(n+1)λ](n+1−A). |
The following discussion can be divided into two cases:
(a) It is easy to see that P(n)>0 if λ<(B−A)+a(1−B)(B−A)(2n−1). To prove that (3.16) holds true, we need only to prove the following inequality
ˆB≤Q(n)P(n)=aA[(n+1)−(A+nB)]B−A+[1−(n+1)λ](n+1−A)a[(n+1)−(A+nB)]B−A+n[1−(n+1)λ]. | (3.17) |
Clearly, Q(n)P(n) is an increasing function with respect to n for a<1,λ<12(n+1) or a>1,λ>12(k+1). If λ<k(B−A)+a(k+1−A−kB)k(k+1)(B−A), then we have
ˆB≤aA[(k+1)−(A+kB)]+[1−(k+1)λ](k+1−A)(B−A)a[(k+1)−(A+kB)]+k[1−(k+1)λ](B−A)=Q(k)P(k)≤Q(n)P(n). |
Therefore, 1a(f1˜△f2)λ(1p,p−1p,a,z)∈¯ΣS∗k(a,A,B) if the conditions (v) or (vi) are satisfied.
(b) It is easy to see that P(n)<0 if λ>(B−A)+a(1−B)(B−A)(2n−1) and P(k)=a[(k+1)−(A+kB)]B−A+k[1−(k+1)λ]>0. To prove that (3.16) holds true, we need only to prove the following inequality
ˆB≤Q(n)P(n)=aA[(n+1)−(A+nB)]A−B−[1−(n+1)λ])(n+1−A)a[(n+1)−(A+nB)]B−A+n[1−(n+1)λ]. | (3.18) |
Clearly, Q(n)P(n) is a decreasing function with respect to n for a<1,λ<12(n+1) or a>1,λ>12(k+1).
ˆB≤limn→+∞Q(n)P(n)≤Q(n)P(n). |
Therefore, 1a(f1˜△f2)λ(1p,p−1p,a,z)∈¯ΣS∗k(a,A,B) if the conditions (vii) or (viii) are satisfied.
To summarize, we conclude that the conclusion (2) is established. Using the same method to that in the proof of (2), the conclusion (1) holds true. Thus, we complete the proof of Theorem 4.
Let A=1 and B=0 in Theorem 4, we have the following corollary.
Corollary 2. Let fi(z)∈¯ΣS∗k(a,1,0)(i=1,2) and p>1. If anyone of the following conditions is satisfied.
(ⅰ) a<1,λ<min{12(n+1),a+1(2n+1),k+a(k+2)k(k+1)},
(ⅱ) a>1,12(k+1)<λ<min{a+1(2n+1),k+a(k+2)k(k+1)},
(ⅲ) a<1,max{a+1(2k+1),k+a(k+2)k(k+1)}<λ<12(n+1),
(ⅳ) a>1,max{12(k+1),a+1(2k+1),k+a(k+2)k(k+1)}<λ.
Then the generalized λ-Hadamard product (f1˜Δf2)λ(1p,p−1p,a;z)∈ ¯ΣS∗k(a,1−2β,ˆB).
This work is supported by the Natural Science Foundation of the People's Republic of China under Grant 11561001, the Natural Science Foundation of Inner Mongolia of the People's Republic of China under Grants 2018MS01026, 2019MS01023 and 2020MS01011, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT-18-A14 and the Higher-School Science Foundation of Inner Mongolia of the People's Republic of China under Grants NJZY20198 and NJZY20200.
All authors declare no conflict of interest in this paper.
[1] | Igoe D, Gavin K, O'Kelly B (2010) Field tests using an instrumented model pipe pile in sand. Physical Modelling in Geotechnics-Proceedings of the 7th International Conference on Physical Modelling in Geotechnics 2010, ICPMG. |
[2] | Igoe D, Doherty P, Gavin K (2010) The development and testing of an instrumented open-ended model pile. Geotech Test J 33: 72–82. |
[3] |
Igoe D, Gavin K, O'Kelly B (2013) An investigation into the use of push-in pile foundations by the offshore wind sector. Int J Environ Stud 70: 777–791. doi: 10.1080/00207233.2013.798496
![]() |
[4] |
Gavin KG, Igoe D, Kirwan L (2013) The effect of ageing on the axial capacity of piles in sand. Proc Inst Civ Eng Geotech Eng 166: 122–130. doi: 10.1680/geng.12.00064
![]() |
[5] | Igoe D, Gavin K, Kirwan L (2013) Investigation into the factors affecting the shaft resistance of driven piles in sands. In: Installation Effects in Geotechnical Engineering-Proceedings of the International Conference on Installation Effects in Geotechnical Engineering, ICIEGE 2013. |
[6] | Murphy G, Igoe D, Doherty P (2018) Gavin KG. 3D FEM approach for laterally loaded monopile design. Comput Geotech 100: 76–83. |
[7] |
Igoe D, Gavin KG, O'Kelly BC (2011) Shaft Capacity of Open-Ended Piles in Sand. J Geotech Geoenvironmental Eng 137: 903–913. doi: 10.1061/(ASCE)GT.1943-5606.0000511
![]() |
[8] | Doherty P, Igoe D (2013) A Driveability Study of Precast Concrete Piles in Dense Sand. DFI J J Deep Found Inst 7: 3–16. |
[9] | Doherty P, Kirwan L, Gavin KG, et al. (2012) Soil Properties at the UCD Geotechnical Research Site at Blessington. In: Bridge and Concrete Research in Ireland, Dublin. |
[10] | Syge FM, West Wicklow (1977) In: South-East Ireland. In: Int Union of Quartenary Research Fieldguide. |
[11] | Cohen JM (1979) Deltaic sedimentation in glacial lake in Blessington in County Wicklow. Moraines Varves, 357–367. |
[12] | Cohen JM (1980) Aspects of glacial lake sedimentation. Trinity College Dublin. |
[13] | Philcox ME (2000) The glacio-lacustrine delta complex at Blessington, Co. Wicklow and related outflow features. In: Guidebook of the 20th Regional Meeting of Sedimentology, Dublin: IAS Special Publication, 129–152. |
[14] | Coxon P (1993) Irish Pleistocene biostratigraphy. Irish J Earth Sci 12: 83–105. |
[15] | Kirwan L (2014) Investigation into Ageing Mechanisms for Axially Loaded Piles Driven in Sand. University College Dublin. |
[16] | Rasband WS (2004) ImageJ. Bethesda, MD: National Institutes of Health. |
[17] | Landini G (2006) Particle8_Plus plugin in ImageJ. |
[18] | Mayne PW (2001) Stress-strain-strength-flow parameters from enhanced in-situ tests. In: Proceedings International Conference on In-Situ Measurement of Soil Properties & Case Histories, Bali, Indonesia, 27–48. |
[19] | Marchetti S, Monaco P, Calabrese M, et al. (2006) Comparison of moduli determined by DMT and back figured from local strain measurements under a 40 m diameter circular test load in the Venice area. Proc From the second international flat dilatometer conference. |
[20] | Spagnoli G, Doherty P, Wu D, et al. (2015) Some mineralogical and geotechnical properties of carbonate and silica sands in relation to a novel mixed-in-place pile, 12th Offshore Mediterranean Conference and Exhibition, Italy: Ravenna. |
[21] | Tolooiyan A and Gavin (2011) Modelling the Cone Penetration Test in sand using Cavity Expansion and Arbitrary Lagrangian Eulerian Finite Element, Computers and Geotechnics, Vol. 38, pp 482-490. |
[22] |
Bolton MD (1986) The strength and dilatancy of sands. Géotechnique 36: 65–78. doi: 10.1680/geot.1986.36.1.65
![]() |
[23] | Kulhawy FH, Mayne PW (1990) Manual on estimating soil properties for foundation design. |
[24] | Robertson PK, Cabal K (2014) Guide to cone penetration testing for geotechnical engineering-6th Edition. |
[25] |
Schnaid F, Yu HS (2007) Interpretation of the seismic cone test in granular soils. Géotechnique 57: 265–272. doi: 10.1680/geot.2007.57.3.265
![]() |
[26] | Gavin K, Jardine RJ, Karlsrud K, et al. (2015) The effects of pile ageing on the shaft capacity of offshore piles in sand. International Symposium Frontiers in Offshore Geotechnics (ISFOG), London: Taylor & Francis, 129–151. |
[27] | Gavin K, Igoe D (2019) A Field Investigation into the Mechanisms of Pile Ageing in Sand, Submitted to Geotechnique. |
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